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The Chemical Connection

  • Xueliang Li
  • Yongtang Shi
  • Ivan Gutman
Chapter

Abstract

Research on what we call the energy of a graph can be traced back to the 1940s or even to the 1930s. In the 1930s, the German scholar Erich Hückel put forward a method for finding approximate solutions of the Schrödinger equation of a class of organic molecules, the so-called conjugated hydrocarbons. Details of this approach, often referred to as the “Hückel molecular orbital (HMO) theory” can be found in appropriate textbooks [76, 101].

Keywords

Molecular Graph Hamiltonian Matrix German Scholar Graph Spectral Theory Orthogonal Basis Function 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 1.
    N. Abreu, D.M. Cardoso, I. Gutman, E.A. Martins, M. Robbiano, Bounds for the signless Laplacian energy. Lin. Algebra Appl. 435, 2365–2374 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    C. Adiga, R. Balakrishnan, W. So, The skew energy of a digraph. Lin. Algebra Appl. 432, 1825–1835 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    C. Adiga, Z. Khoshbakht, I. Gutman, More graphs whose energy exceeds the number of vertices. Iran. J. Math. Sci. Inf. 2(2), 13–19 (2007)Google Scholar
  4. 4.
    C. Adiga, M. Smitha. On the skew Laplacian energy of a digraph. Int. Math. Forum 4, 1907–1914 (2009)MathSciNetzbMATHGoogle Scholar
  5. 5.
    C. Adiga, M. Smitha, On maximum degree energy of a graph. Int. J. Contemp. Math. Sci. 4, 385–396 (2009)MathSciNetzbMATHGoogle Scholar
  6. 6.
    J. Aihara, A new definition of Dewar-type resonance energies. J. Am. Chem. Soc. 98, 2750–2758 (1976)CrossRefGoogle Scholar
  7. 7.
    AIM Workshop on Spectra of Families of Matrices Described by Graphs, Digraphs, and Sign Patterns – Open Questions, 7 December 2006Google Scholar
  8. 8.
    S. Akbari, E. Ghorbani, Choice number and energy of graphs. Lin. Algebra Appl. 429, 2687–2690 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    S. Akbari, E. Ghorbani, J.H. Koolen, M.R. Oboudi, On sum of powers of the Laplacian and signless Laplacian eigenvalues of graphs. Electron. J. Combinator. 17, R115 (2010)MathSciNetGoogle Scholar
  10. 10.
    S. Akbari, E. Ghorbani, M.R. Oboudi, Edge addition, singular values and energy of graphs and matrices. Lin. Algebra Appl. 430, 2192–2199 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    S. Akbari, E. Ghorbani, S. Zare, Some relations between rank, chromatic number and energy of graphs. Discr. Math. 309, 601–605 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    S. Akbari, F. Moazami, S. Zare, Kneser graphs and their complements are hyperenergetic. MATCH Commun. Math. Comput. Chem. 61, 361–368 (2009)MathSciNetzbMATHGoogle Scholar
  13. 13.
    T. Aleksić, Upper bounds for Laplacian energy of graphs. MATCH Commun. Math. Comput. Chem. 60, 435–439 (2008)MathSciNetzbMATHGoogle Scholar
  14. 14.
    F. Alinaghipour, B. Ahmadi, On the energy of complement of regular line graph. MATCH Commun. Math. Comput. Chem. 60, 427–434 (2008)MathSciNetzbMATHGoogle Scholar
  15. 15.
    A. Alwardi, N.D. Soner, I. Gutman, On the common-neighborhood energy of a graph. Bull. Acad. Serbe Sci. Arts (Cl. Math. Nat.) 143, 49–59 (2011)Google Scholar
  16. 16.
    E.O.D. Andriantiana, Unicyclic bipartite graphs with maximum energy. MATCH Commun. Math. Comput. Chem. 66, 913–926 (2011)MathSciNetGoogle Scholar
  17. 17.
    E.O.D. Andriantiana, More trees with large energy. MATCH Commun. Math. Comput. Chem. 68, 675–695 (2012)Google Scholar
  18. 18.
    E.O.D. Andriantiana, S. Wagner, Unicyclic graphs with large energy. Lin. Algebra Appl. 435, 1399–1414 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    G. Anderson, O. Zeitouni, A CLT for a band matrix model. Probab. Theor. Relat. Field. 134, 283–338 (2005)MathSciNetCrossRefGoogle Scholar
  20. 20.
    M. Aouchiche, P. Hansen, A survey of automated conjectures in spectral graph theory. Lin. Algebra Appl. 432, 2293–2322 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    M. Aouchiche, P. Hansen, A survey of Nordhaus–Gaddum type relations. Les Cahiers du GERAD G-2010-74, X+1–81 (2010)Google Scholar
  22. 22.
    S.K. Ayyaswamy, S. Balachandran, I. Gutman, On second-stage spectrum and energy of a graph. Kragujevac J. Math. 34, 139–146 (2010)MathSciNetGoogle Scholar
  23. 23.
    S.K. Ayyaswamy, S. Balachandran, I. Gutman, Upper bound for the energy of strongly connected digraphs. Appl. Anal. Discr. Math. 5, 37–45 (2011)MathSciNetCrossRefGoogle Scholar
  24. 24.
    D. Babić, I. Gutman, More lower bounds for the total π-electron energy of alternant hydrocarbons. MATCH Commun. Math. Comput. Chem. 32, 7–17 (1995)Google Scholar
  25. 25.
    Z.D. Bai, Methodologies in spectral analysis of large dimensional random matrices, a review, Statistica Sinica 9, 611–677 (1999)MathSciNetzbMATHGoogle Scholar
  26. 26.
    R. Balakrishnan, The energy of a graph. Lin. Algebra Appl. 387, 287–295 (2004)zbMATHCrossRefGoogle Scholar
  27. 27.
    R.B. Bapat, Graphs and Matrices, Section 3.4 (Springer, Hindustan Book Agency, London, 2011)Google Scholar
  28. 28.
    R.B. Bapat, S. Pati, Energy of a graph is never an odd integer. Bull. Kerala Math. Assoc. 1, 129–132 (2004)MathSciNetGoogle Scholar
  29. 29.
    A. Barenstein, R. Gay, Complex Variables (Springer, New York, 1991)CrossRefGoogle Scholar
  30. 30.
    S. Barnard, J.M. Child, Higher Algebra (MacMillan, London, 1952)Google Scholar
  31. 31.
    R. Bhatia, Matrix Analysis (Springer, New York, 1997)CrossRefGoogle Scholar
  32. 32.
    F.M. Bhatti, K.C. Das, S.A. Ahmed, On the energy and spectral properties of the He matrix of the hexagonal systems. Czech. Math. J., in pressGoogle Scholar
  33. 33.
    N. Biggs, Algebriac Graph Theory (Cambridge University Press, Cambridge, 1993)Google Scholar
  34. 34.
    P. Billingsley, Probability and Measure (Wiley, New York, 1995)zbMATHGoogle Scholar
  35. 35.
    S.R. Blackburn, I.E. Shparlinski, On the average energy of circulant graphs. Lin. Algebra Appl. 428, 1956–1963 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  36. 36.
    D.A. Bochvar, I.V. Stankevich, Approximate formulas for some characteristics of the electron structure of molecules, 1. Total electron energy. Zh. Strukt. Khim. 21, 61–66 (in Russian) (1980)Google Scholar
  37. 37.
    B. Bollobás, Extremal Graph Theory (Academic, London, 1978)zbMATHGoogle Scholar
  38. 38.
    B. Bollobás, Random Graphs (Cambridge University Press, Cambridge, 2001)zbMATHCrossRefGoogle Scholar
  39. 39.
    J.A. Bondy, U.S.R. Murty, Graph Theory with Applications (MacMllan, London, 1976)zbMATHGoogle Scholar
  40. 40.
    J.A. Bondy, U.S.R. Murty, Graph Theory (Springer, Berlin, 2008)zbMATHCrossRefGoogle Scholar
  41. 41.
    A.S. Bonifácio, N.M.M. de Abreu, C.T.M. Vinagre, I. Gutman, Hyperenergetic and non-hyperenergetic graphs, in Proceedings of the XXXI Congresso Nacional de Matematica Applicada e Computacional (CNMAC 2008), Belem (Brazil), 2008, pp. 1–6 (in Portuguese)Google Scholar
  42. 42.
    A.S. Bonifácio, C.T.M. Vinagre, N.M.M. de Abreu, Constructing pairs of equienergetic and non-cospectral graphs. Appl. Math. Lett. 21, 338–341 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  43. 43.
    B. Borovićanin, I. Gutman, in Nullity of Graphs, ed. by D. Cvetković, I. Gutman. Applications of Graph Spectra (Mathematical Institute, Belgrade, 2009), pp. 107–122Google Scholar
  44. 44.
    S.B. Bozkurt, A.D. Güngör, I. Gutman, A.S. Çevik, Randić matrix and Randić energy. MATCH Commun. Math. Comput. Chem. 64, 239–250 (2010)MathSciNetGoogle Scholar
  45. 45.
    S.B. Bozkurt, A.D. Güngör, B. Zhou, Note on the distance energy of graphs. MATCH Commun. Math. Comput. Chem. 64, 129–134 (2010)MathSciNetGoogle Scholar
  46. 46.
    V. Božin, M. Mateljević, Energy of Graphs and Orthogonal Matrices, ed. by W. Gautschi, G. Mastroianni, T.M. Rassias. Approximation and Computation – In Honor of Gradimir V. Milovanović (Springer, New York, 2011), pp. 85–94Google Scholar
  47. 47.
    V. Brankov, D. Stevanović, I. Gutman, Equienergetic chemical trees. J. Serb. Chem. Soc. 69, 549–553 (2004)CrossRefGoogle Scholar
  48. 48.
    A.E. Brouwer, A.M. Cohen, A. Neumaier, Distance–Regular Graphs (Springer, New York, 1989)zbMATHGoogle Scholar
  49. 49.
    A.E. Brouwer, W.H. Haemers, Spectra of Graphs (Springer, Berlin, 2012)zbMATHCrossRefGoogle Scholar
  50. 50.
    R. Brualdi, Energy of a Graph, in: Notes for AIM Workshop on Spectra of Families of Matrices Described by Graphs, Digraphs, and Sign Patterns, 2006Google Scholar
  51. 51.
    Y. Cao, A. Lin, R. Luo, X. Zha, On the minimal energy of unicyclic Hückel molecular graphs possessing Kekulé structures. Discr. Appl. Math. 157, 913–919 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  52. 52.
    G. Caporossi, E. Chasset, B. Furtula, Some conjectures and properties on distance energy. Les Cahiers du GERAD G-2009-64, V + 1–7 (2009)Google Scholar
  53. 53.
    G. Caporossi, D. Cvetković, I. Gutman, P. Hansen, Variable neighborhood search for extremal graphs. 2. Finding graphs with extremal energy. J. Chem. Inf. Comput. Sci. 39, 984–996 (1999)Google Scholar
  54. 54.
    D.M. Cardoso, E.A. Martins, M. Robbiano, V. Trevisan, Computing the Laplacian spectra of some graphs. Discr. Appl. Math. doi:10.1016/j.dam.2011.04.002Google Scholar
  55. 55.
    D.M. Cardoso, I. Gutman, E.A. Martins, M. Robbiano, A generalization of Fiedler’s lemma and some applications. Lin. Multilin. Algebra 435, 2365–2374 (2011)MathSciNetzbMATHGoogle Scholar
  56. 56.
    P.C. Carter, An empirical equation for the resonance energy of polycyclic aromatic hydrocarbons. Trans. Faraday Soc. 45, 597–602 (1949)CrossRefGoogle Scholar
  57. 57.
    M. Cavers, S. Fallat, S. Kirkland, On the normalized Laplacian energy and general Randić index R  − 1 of graphs. Lin. Algebra Appl. 433, 172–190 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  58. 58.
    A. Chen, A. Chang, W.C. Shiu, Energy ordering of unicyclic graphs. MATCH Commun. Math. Comput. Chem. 55, 95–102 (2006)MathSciNetzbMATHGoogle Scholar
  59. 59.
    B. Cheng, B. Liu, On the nullity of graphs. El. J. Lin. Algebra 16, 60–67 (2007)MathSciNetzbMATHGoogle Scholar
  60. 60.
    C.M. Cheng, R.A. Horn, C.K. Li, Inequalities and equalities for the Cartesian decomposition of complex matrices. Lin. Algebra Appl. 341, 219–237 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  61. 61.
    R. Churchill, J. Brown, Complex Variables and Applications (McGraw–Hill, New York, 1984)Google Scholar
  62. 62.
    J. Cioslowski, Upper bound for total π-electron energy of benzenoid hydrocarbons. Z. Naturforsch. 40a, 1167–1168 (1985)Google Scholar
  63. 63.
    J. Cioslowski, The use of the Gauss–Chebyshev quadrature in estimation of the total π-electron energy of benzenoid hydrocarbons. Z. Naturforsch. 40a, 1169–1170 (1985)Google Scholar
  64. 64.
    J. Cioslowski, Additive nodal increments for approximate calculation of the total π-electron energy of benzenoid hydrocarbons. Theor. Chim. Acta 68, 315–319 (1985)CrossRefGoogle Scholar
  65. 65.
    J. Cioslowski, Decomposition of the total π-electron energy of polycyclic hydrocarbons into the benzene ring increments. Chem. Phys. Lett. 122, 234–236 (1985)CrossRefGoogle Scholar
  66. 66.
    J. Cioslowski, The generalized McClelland formula. MATCH Commun. Math. Chem. 20, 95–101 (1986)Google Scholar
  67. 67.
    J. Cioslowski, A unified theory of the stability of benzenoid hydrocarbons. Int. J. Quantum Chem. 31, 581–590 (1987)CrossRefGoogle Scholar
  68. 68.
    J. Cioslowski, Scaling properties of topological invariants. Topics Curr. Chem. 153, 85–99 (1990)CrossRefGoogle Scholar
  69. 69.
    J. Cioslowski, A final solution of the problem concerning the (N, M, K)-dependence of the total π-electron energy of conjugated systems? MATCH Commun. Math. Chem. 25, 83–93 (1990)Google Scholar
  70. 70.
    J. Cioslowski, I. Gutman, Upper bounds for the total π-electron energy of benzenoid hydrocarbons and their relations. Z. Naturforsch. 41a, 861–865 (1986)Google Scholar
  71. 71.
    V. Consonni, R. Todeschini, New spectral index for molecule description. MATCH Commun. Math. Comput. Chem. 60, 3–14 (2008)MathSciNetzbMATHGoogle Scholar
  72. 72.
    J. Conway, Functions of One Complex Variable (Springer, Berlin, 1978)CrossRefGoogle Scholar
  73. 73.
    C.A. Coulson, On the calculation of the energy in unsaturated hydrocarbon molecules. Proc. Cambridge Phil. Soc. 36, 201–203 (1940)CrossRefGoogle Scholar
  74. 74.
    C.A. Coulson, J. Jacobs, Conjugation across a single bond. J. Chem. Soc. 2805–2812 (1949)Google Scholar
  75. 75.
    C.A. Coulson, H.C. Longuet–Higgins, The electronic structure of conjugated systems. I. General theory. Proc. Roy. Soc. A 191, 39–60 (1947)Google Scholar
  76. 76.
    C.A. Coulson, B. O’Leary, R.B. Mallion, Hückel Theory for Organic Chemists (Academic, London, 1978)Google Scholar
  77. 77.
    R. Craigen, H. Kharaghani, in Hadamard Matrices and Hadamard Designs, ed. by C.J. Colbourn, J.H. Denitz. Handbook of Combinatorial Designs, Chapter V.1 (Chapman & Hall/CRC, Boca Raton, 2007)Google Scholar
  78. 78.
    Z. Cui, B. Liu, On Harary matrix, Harary index and Harary energy. MATCH Commun. Math. Comput. Chem. 68, 815–823 (2012)Google Scholar
  79. 79.
    D. Cvetković, A table of connected graphs on six vertices. Discr. Math. 50, 37–49 (1984)zbMATHCrossRefGoogle Scholar
  80. 80.
    D. Cvetković, M. Doob, I. Gutman, A. Torgašev, Recent Results in the Theory of Graph Spectra (North–Holland, Amsterdam, 1988)Google Scholar
  81. 81.
    D. Cvetković, M. Doob, H. Sachs, Spectra of Graphs – Theory and Application (Academic, New York, 1980)Google Scholar
  82. 82.
    D. Cvetković, J. Grout, Graphs with extremal energy should have a small number of distinct eigenvalues. Bull. Acad. Serbe Sci. Arts (Cl. Math. Natur.) 134, 43–57 (2007)Google Scholar
  83. 83.
    D. Cvetković, I. Gutman, The algebraic multiplicity of the number zero in the spectrum of a bipartite graph. Mat. Vesnik, 9, 141–150 (1972)MathSciNetGoogle Scholar
  84. 84.
    D. Cvetković, I. Gutman, The computer system GRAPH: A useful tool in chemical graph theory. J. Comput. Chem. 7, 640–644 (1986)CrossRefGoogle Scholar
  85. 85.
    D. Cvetković, I. Gutman (eds.), Applications of Graph Spectra (Mathematical Institution, Belgrade, 2009)zbMATHGoogle Scholar
  86. 86.
    D. Cvetković, I. Gutman (eds.) Selected Topics on Applications of Graph Spectra (Mathematical Institute, Belgrade, 2011)zbMATHGoogle Scholar
  87. 87.
    D. Cvetković, M. Petrić, A table of connected graphs on six vertices. Discr. Math. 50, 37–49 (1984)zbMATHCrossRefGoogle Scholar
  88. 88.
    D. Cvetković, P. Rowlinson, S. Simić, Signless Laplacians of finite graphs. Lin. Algebra Appl. 423, 155–171 (2007)zbMATHCrossRefGoogle Scholar
  89. 89.
    D. Cvetković, P. Rowlinson, S. Simić, An Introduction to the Theory of Graph Spectra. (Cambridge University Press, Cambridge, 2010)Google Scholar
  90. 90.
    K.C. Das, Sharp bounds for the sum of the squares of the degrees of a graph. Kragujevac J. Math. 25, 31–49 (2003)MathSciNetzbMATHGoogle Scholar
  91. 91.
    K.C. Das, Maximizing the sum of the squares of the degrees of a graph. Discr. Math. 285, 57–66 (2004)zbMATHCrossRefGoogle Scholar
  92. 92.
    K.C. Das, F.M. Bhatti, S.G. Lee, I. Gutman, Spectral properties of the He matrix of hexagonal systems. MATCH Commun. Math. Comput. Chem. 65, 753–774 (2011)MathSciNetGoogle Scholar
  93. 93.
    K.C. Das, P. Kumar, Bounds on the greatest eigenvalue of graphs. Indian J. Pure Appl. Math. 34, 917–925 (2003)MathSciNetzbMATHGoogle Scholar
  94. 94.
    J. Day, W. So, Singular value inequality and graph energy change. El. J. Lin. Algebra 16, 291–299 (2007)MathSciNetzbMATHGoogle Scholar
  95. 95.
    J. Day, W. So, Graph energy change due to edge deletion. Lin. Algebra Appl. 428, 2070–2078 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  96. 96.
    N.N.M. de Abreu, C.T.M. Vinagre, A.S. Bonifácio, I. Gutman, The Laplacian energy of some Laplacian integral graphs. MATCH Commun. Math. Comput. Chem. 60, 447–460 (2008)MathSciNetzbMATHGoogle Scholar
  97. 97.
    D. de Caen, An upper bound on the sum of squares of degrees in a graph. Discr. Math. 185, 245–248 (1998)zbMATHCrossRefGoogle Scholar
  98. 98.
    J.A. de la Peña, L. Mendoza, Moments and π-electron energy of hexagonal systems in 3-space. MATCH Commun. Math. Comput. Chem. 56, 113–129 (2006)MathSciNetzbMATHGoogle Scholar
  99. 99.
    J.A. de la Peña, L. Mendoza, J. Rada, Comparing momenta and π-electron energy of benzenoid molecules. Discr. Math. 302, 77–84 (2005)zbMATHCrossRefGoogle Scholar
  100. 100.
    P. Deift, Orthogonal Polynomials and Random Matrices – A Riemann–Hilbert Approach (American Mathematical Society, New York, 2000)zbMATHGoogle Scholar
  101. 101.
    M.J.S. Dewar, The Molecular Orbital Theory of Organic Chemistry (McGraw–Hill, New York, 1969)Google Scholar
  102. 102.
    M. Doob, Graphs with a small number of distinct eigenvalues. Ann. New York Acad. Sci. 175, 104–110 (1970)MathSciNetzbMATHGoogle Scholar
  103. 103.
    W. Du, X. Li, Y. Li, Various energies of random graphs. MATCH Commun. Math. Comput. Chem. 64, 251–260 (2010)MathSciNetGoogle Scholar
  104. 104.
    W. Du, X. Li, Y. Li, The Laplacian energy of random graphs. J. Math. Anal. Appl. 368, 311–319 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  105. 105.
    W. Du, X. Li, Y. Li, The energy of random graphs. Lin. Algebra Appl. 435, 2334–2346 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  106. 106.
    W. England, K. Ruedenberg, Why is the delocalization energy negative and why is it proportional to the number of π electrons? J. Am. Chem. Soc. 95, 8769–8775 (1973)CrossRefGoogle Scholar
  107. 107.
    S. Fajtlowicz, On conjectures of Grafitti. II. Congr. Numer. 60, 187–197 (1987)MathSciNetGoogle Scholar
  108. 108.
    K. Fan, Maximum properties and inequalities for the eigenvalues of completely continuous operators. Proc. Natl. Acad. Sci. USA 37, 760–766 (1951)zbMATHCrossRefGoogle Scholar
  109. 109.
    G.H. Fath-Tabar, A.R. Ashrafi, Some remarks on Laplacian eigenvalues and Laplacian energy of graphs. Math. Commun. 15, 443–451 (2010)MathSciNetzbMATHGoogle Scholar
  110. 110.
    G.H. Fath-Tabar, A.R. Ashrafi, I. Gutman, Note on Laplacian energy of graphs. Bull. Acad. Serbe Sci. Arts (Cl. Math. Natur.) 137, 1–10 (2008)Google Scholar
  111. 111.
    E.J. Farrell, An introduction to matching polynomials. J. Comb. Theor. B 27, 75–86 (1979)zbMATHCrossRefGoogle Scholar
  112. 112.
    E.J. Farrell, The matching polynomial and its relation to the acyclic polynomial of a graph. Ars Combin. 9, 221–228 (1980)MathSciNetzbMATHGoogle Scholar
  113. 113.
    O. Favaron, M. Mahéo, J.F. Saclé, Some eigenvalue properties of graphs (Conjectures of Grafitti – II). Discr. Math. 111, 197–220 (1993)zbMATHCrossRefGoogle Scholar
  114. 114.
    M. Fiedler, Additive compound matrices and an inequality for eigenvalues of symmetric stochastic matrices. Czech. Math. J. 24, 392–402 (1974)MathSciNetGoogle Scholar
  115. 115.
    M. Fiedler, Additive compound graphs. Discr. Math. 187, 97–108 (1998)MathSciNetzbMATHGoogle Scholar
  116. 116.
    S. Fiorini, I. Gutman, I. Sciriha, Trees with maximum nullity. Lin. Algebra Appl. 397, 245–251 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  117. 117.
    P.W. Fowler, Energies of Graphs and Molecules, ed. by T.E. Simos, G. Maroulis. Computational Methods in Modern Science and Engineering, vol. 2 (Springer, New York, 2010), pp. 517–520Google Scholar
  118. 118.
    H. Fripertinger, I. Gutman, A. Kerber, A. Kohnert, D. Vidović, The energy of a graph and its size dependence. An improved Monte Carlo approach. Z. Naturforsch. 56a, 342–346 (2001)Google Scholar
  119. 119.
    E. Fritscher, C. Hoppen, I. Rocha, V. Trevisan, On the sum of the Laplacian eigenvalues of a tree. Lin. Algebra Appl. 435, 371–399 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  120. 120.
    Z. Füredi, J. Komlós, The eigenvalues of random symmetric matrices. Combinatorica 1, 233–241 (1981)MathSciNetzbMATHCrossRefGoogle Scholar
  121. 121.
    B. Furtula, S. Radenković, I. Gutman, Bicyclic molecular graphs with greatest energy. J. Serb. Chem. Soc. 73, 431–433 (2008)CrossRefGoogle Scholar
  122. 122.
    K.A. Germina, S.K. Hameed, T. Zaslavsky, On products and line graphs of signed graphs, their eigenvalues and energy. Lin. Algebra Appl. 435, 2432–2450 (2011)zbMATHCrossRefGoogle Scholar
  123. 123.
    E. Ghorbani, J.H. Koolen, J.Y. Yang, Bounds for the Hückel energy of a graph. El. J. Comb. 16, #R134 (2009)Google Scholar
  124. 124.
    C.D. Godsil, I. Gutman, On the theory of the matching polynomial. J. Graph Theor. 5, 137–144 (1981)MathSciNetzbMATHCrossRefGoogle Scholar
  125. 125.
    C. Godsil, G. Royle, Algebraic Graph Theory (Springer, New York, 2001)zbMATHCrossRefGoogle Scholar
  126. 126.
    S.C. Gong, G.H. Xu, 3-Regular digraphs with optimum skew energy. Lin. Algebra Appl. 436, 465–471 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  127. 127.
    A. Graovac, D. Babić, K. Kovačević, Simple estimates of the total and the reference pi-electron energy of conjugated hydrocarbons. Stud. Phys. Theor. Chem. 51, 448–457 (1987)Google Scholar
  128. 128.
    A. Graovac, I. Gutman, P.E. John, D. Vidović, I. Vlah, On statistics of graph energy. Z. Naturforsch. 56a, 307–311 (2001)Google Scholar
  129. 129.
    A. Graovac, I. Gutman, O.E. Polansky, Topological effect on MO energies, IV. The total π-electron energy of S– and T-isomers. Monatsh. Chem. 115, 1–13 (1984)Google Scholar
  130. 130.
    A. Graovac, I. Gutman, N. Trinajstić, On the Coulson integral formula for total π-electron energy. Chem. Phys. Lett. 35, 555–557 (1975)CrossRefGoogle Scholar
  131. 131.
    A. Graovac, I. Gutman, N. Trinajstić, A linear relationship between the total π-electron energy and the characteristic polynomial. Chem. Phys. Lett. 37, 471–474 (1976)CrossRefGoogle Scholar
  132. 132.
    A. Graovac, I. Gutman, N. Trinajstić, Graph–theoretical study of conjugated hydrocarbons: Total pi-electron energies and their differences. Int. J. Quantum Chem. 12(Suppl. 1), 153–155 (1977)Google Scholar
  133. 133.
    A. Graovac, I. Gutman, N. Trinajstić, Topological Approach to the Chemistry of Conjugated Molecules (Springer, Berlin, 1977)zbMATHCrossRefGoogle Scholar
  134. 134.
    R. Grone, R. Merris, The Laplacian spectrum of a graph II. SIAM J. Discr. Math. 7, 221–229 (1994)MathSciNetzbMATHCrossRefGoogle Scholar
  135. 135.
    R. Grone, R. Merris, V.S. Sunder, The Laplacian spectrum of a graph. SIAM J. Matrix Anal. Appl. 11, 218–238 (1990)MathSciNetzbMATHCrossRefGoogle Scholar
  136. 136.
    E. Gudiño, J. Rada, A lower bound for the spectral radius of a digraph. Lin. Algebra Appl. 433, 233–240 (2010)zbMATHCrossRefGoogle Scholar
  137. 137.
    A.D. Güngör, S.B. Bozkurt, On the distance spectral radius and distance energy of graphs. Lin. Multilin. Algebra 59, 365–370 (2011)zbMATHCrossRefGoogle Scholar
  138. 138.
    A.D. Güngör, A.S. Çevik, On the Harary energy and Harary Estrada index of a graph. MATCH Commun. Math. Comput. Chem. 64, 281–296 (2010)MathSciNetGoogle Scholar
  139. 139.
    H.H. Günthard, H. Primas, Zusammenhang von Graphentheorie und MO–Theorie von Molekeln mit Systemen konjugierter Bindungen. Helv. Chim. Acta 39, 1645–1653 (1956)CrossRefGoogle Scholar
  140. 140.
    J. Guo, Sharp upper bounds for total π-electron energy of alternant hydrocarbons. J. Math. Chem. 43, 713–718 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  141. 141.
    J. Guo, On the minimal energy ordering of trees with perfect matchings. Discr. Appl. Math. 156, 2598–2605 (2008)zbMATHCrossRefGoogle Scholar
  142. 142.
    I. Gutman, Bounds for total π-electron energy. Chem. Phys. Lett. 24, 283–285 (1974)CrossRefGoogle Scholar
  143. 143.
    I. Gutman, Estimating the π-electron energy of very large conjugated systems. Die Naturwissenschaften 61, 216–217 (1974)CrossRefGoogle Scholar
  144. 144.
    I. Gutman, The nonexistence of topological formula for total π-electron energy. Theor. Chim. Acta 35, 355–359 (1974)CrossRefGoogle Scholar
  145. 145.
    I. Gutman, Acyclic systems with extremal Hückel π-electron energy. Theor. Chim. Acta 45, 79–87 (1977)CrossRefGoogle Scholar
  146. 146.
    I. Gutman, Bounds for total π-electron energy of polymethines. Chem. Phys. Lett. 50, 488–490 (1977)CrossRefGoogle Scholar
  147. 147.
    I. Gutman, A class of approximate topological formulas for total π-electron energy. J. Chem. Phys. 66, 1652–1655 (1977)CrossRefGoogle Scholar
  148. 148.
    I. Gutman, A topological formula for total π-electron energy. Z. Naturforsch. 32a, 1072–1073 (1977)Google Scholar
  149. 149.
    I. Gutman, The energy of a graph. Ber. Math.–Statist. Sekt. Forschungsz. Graz 103, 1–22 (1978)Google Scholar
  150. 150.
    I. Gutman, Bounds for Hückel total π-electron energy. Croat. Chem. Acta 51, 299–306 (1978)Google Scholar
  151. 151.
    I. Gutman, The matching polynomial. MATCH Commun. Math. Comput. Chem. 6, 75–91 (1979)zbMATHGoogle Scholar
  152. 152.
    I. Gutman, Total π-electron energy of a class of conjugated polymers. Bull. Soc. Chim. Beograd 45, 67–68 (1980)Google Scholar
  153. 153.
    I. Gutman, New approach to the McClelland approximation. MATCH Commun. Math. Comput. Chem. 14, 71–81 (1983)Google Scholar
  154. 154.
    I. Gutman, Bounds for total π-electron energy of conjugated hydrocarbons. Z. Phys. Chem. (Leipzig) 266, 59–64 (1985)Google Scholar
  155. 155.
    I. Gutman, Acyclic conjugated molecules, tree and their energies. J. Math. Chem. 1, 123–143 (1987)MathSciNetGoogle Scholar
  156. 156.
    I. Gutman, The generalized Cioslowski formula. MATCH Commun. Math. Comput. Chem. 22, 269–275 (1987)Google Scholar
  157. 157.
    I. Gutman, On the dependence of the total π-electron energy of a benzenoid hydrocarbon on the number of Kekulé structures. Chem. Phys. Lett. 156, 119–121 (1989)CrossRefGoogle Scholar
  158. 158.
    I. Gutman, McClelland-type lower bound for total π-electron energy. J. Chem. Soc. Faraday Trans. 86, 3373–3375 (1990)CrossRefGoogle Scholar
  159. 159.
    I. Gutman, McClelland–type approximations for total π-electron energy of benzenoid hydrocarbons. MATCH Commun. Math. Comput. Chem. 26, 123–135 (1991)Google Scholar
  160. 160.
    I. Gutman, Estimation of the total π-electron energy of a conjugated molecule. J. Chin. Chem. Soc. 39, 1–5 (1992)Google Scholar
  161. 161.
    I. Gutman, Total π-electron energy of benzenoid hydrocarbons. Topics Curr. Chem. 162, 29–63 (1992)CrossRefGoogle Scholar
  162. 162.
    I. Gutman, Remark on the moment expansion of total π-electron energy. Theor. Chim. Acta 83, 313–318 (1992)CrossRefGoogle Scholar
  163. 163.
    I. Gutman, Approximating the total π-electron energy of benzenoid hydrocarbons: A record accurate formula of (n, m)-type. MATCH Commun. Math. Comput. Chem. 29, 61–69 (1993)zbMATHGoogle Scholar
  164. 164.
    I. Gutman, Approximating the total π-electron energy of benzenoid hydrocarbons: On an overlooked formula of Cioslowski. MATCH Commun. Math. Comput. Chem. 29, 71–79 (1993)zbMATHGoogle Scholar
  165. 165.
    I. Gutman, A regularity for the total π-electron energy of phenylenes. MATCH Commun. Math. Comput. Chem. 31, 99–110 (1994)Google Scholar
  166. 166.
    I. Gutman, An approximate Hückel total π-electron energy formula for benzenoid aromatics: Some amendments. Polyc. Arom. Comp. 4, 271–274 (1995)CrossRefGoogle Scholar
  167. 167.
    I. Gutman, A class of lower bounds for total π-electron energy of alternant conjugated hydrocarbons. Croat. Chem. Acta 68, 187–192 (1995)Google Scholar
  168. 168.
    I. Gutman, On the energy of quadrangle-free graphs. Coll. Sci. Papers Fac. Sci. Kragujevac 18, 75–82 (1996)zbMATHGoogle Scholar
  169. 169.
    I. Gutman, Note on Türker’s approximate formula for total π-electron energy of benzenoid hydrocarbons. ACH – Models Chem. 133, 415–420 (1996)Google Scholar
  170. 170.
    I. Gutman, Hyperenergetic molecular graphs. J. Serb. Chem. Soc. 64, 199–205 (1999)Google Scholar
  171. 171.
    I. Gutman, On the Hall rule in the theory of benzenoid hydrocarbons. Int. J. Quant. Chem. 74, 627–632 (1999)CrossRefGoogle Scholar
  172. 172.
    I. Gutman, A simple (n, m)-type estimate of the total π-electron energy. Indian J. Chem. 40A, 929–932 (2001)Google Scholar
  173. 173.
    I. Gutman, in The Energy of a Graph: Old and New Results, ed. by A. Betten, A. Kohnert, R. Laue, A. Wassermann. Algebraic Combinatorics and Applications (Springer, Berlin, 2001), pp. 196–211Google Scholar
  174. 174.
    I. Gutman, Topology and stability of conjugated hydrocarbons. The dependence of total π-electron energy on moleculr topology. J. Serb. Chem. Soc. 70, 441–456 (2005)Google Scholar
  175. 175.
    I. Gutman, Cyclic conjugation energy effects in polycyclic π-electron systems. Monatsh. Chem. 136, 1055–1069 (2005)CrossRefGoogle Scholar
  176. 176.
    I. Gutman, in Chemical Graph Theory – The Mathematical Connection, ed. by J.R. Sabin, E.J. Brändas. Advances in Quantum Chemistry 51 (Elsevier, Amsterdam, 2006), pp. 125–138Google Scholar
  177. 177.
    I. Gutman, On graphs whose energy exceeds the number of vertices. Lin. Algebra Appl. 429, 2670–2677 (2008)zbMATHCrossRefGoogle Scholar
  178. 178.
    I. Gutman, in Hyperenergetic and Hypoenergetic Graphs, ed. by D. Cvetković, I. Gutman. Selected Topics on Applications of Graph Spectra (Mathematical Institute, Belgrade, 2011), pp. 113–135Google Scholar
  179. 179.
    I. Gutman, Generalizing the McClelland and Koolen–Moulton inequalities for total π-electron energy. Int. J. Chem. Model. 3, (2012) in pressGoogle Scholar
  180. 180.
    I. Gutman, A.R. Ashrafi, G.H. Fath–Tabar, Equienergetic graphs. Farhang va Andishe-e-Riazi 15, 41–50 (1389) (in Persian, 1389 ∼ 2011)Google Scholar
  181. 181.
    I. Gutman, N. Cmiljanović, S. Milosavljević, S. Radenković, Effect of non-bonding molecular orbitals on total π-electron energy. Chem. Phys. Lett. 383, 171–175 (2004)CrossRefGoogle Scholar
  182. 182.
    I. Gutman, N. Cmiljanović, S. Milosavljević, S. Radenković, Dependence of total π-electron energy on the number of non-bonding molecular orbitals. Monatsh. Chem. 135, 765–772 (2004)CrossRefGoogle Scholar
  183. 183.
    I. Gutman, S.J. Cyvin, Introduction to the Theory of Benzenoid Hydrocarbons (Springer, Berlin, 1989)CrossRefGoogle Scholar
  184. 184.
    I. Gutman, N.M.M. de Abreu, C.T.M. Vinagre, A.S. Bonifácio, S. Radenković, Relation between energy and Laplacian energy. MATCH Commun. Math. Comput. Chem. 59, 343–354 (2008)MathSciNetzbMATHGoogle Scholar
  185. 185.
    I. Gutman, B. Furtula, H. Hua, Bipartite unicyclic graphs with maximal, second-maximal, and third-maximal energy. MATCH Commun. Math. Comput. Chem. 58, 85–92 (2007)MathSciNetGoogle Scholar
  186. 186.
    I. Gutman, B. Furtula, D. Vidović, Coulson function and total π-electron energy. Kragujevac J. Sci. 24, 71–82 (2002)Google Scholar
  187. 187.
    I. Gutman, A. Graovac, S. Vuković, S. Marković, Some more isomer-undistinguishing approximate formulas for the total π-electron energy of benzenoid hydrocarbons. J. Serb. Chem. Soc. 54, 189–196 (1989)Google Scholar
  188. 188.
    I. Gutman, E. Gudiño, D. Quiroz, Upper bound for the energy of graphs with fixed second and fourth spectral moments. Kragujevac J. Math. 32, 27–35 (2009)MathSciNetzbMATHGoogle Scholar
  189. 189.
    I. Gutman, G.G. Hall, Linear dependence of total π-electron energy of benzenoid hydrocarbons on Kekulé structure count. Int. J. Quant. Chem. 41, 667–672 (1992)CrossRefGoogle Scholar
  190. 190.
    I. Gutman, G.G. Hall, S. Marković, Z. Stanković, V. Radivojević, Effect of bay regions on the total π-electron energy of benzenoid hydrocarbons. Polyc. Arom. Comp. 2, 275–282 (1991)CrossRefGoogle Scholar
  191. 191.
    I. Gutman, Y. Hou, Bipartite unicyclic graphs with greatest energy. MATCH Commun. Math. Comput. Chem. 43, 17–28 (2001)MathSciNetzbMATHGoogle Scholar
  192. 192.
    I. Gutman, Y. Hou, H.B. Walikar, H.S. Ramane, P.R. Hampiholi, No Hückel graph is hyperenergetic. J. Serb. Chem. Soc. 65, 799–801 (2000)Google Scholar
  193. 193.
    I. Gutman, G. Indulal, R. Todeschini, Generalizing the McClelland bounds for total π-electron energy. Z. Naturforsch. 63a, 280–282 (2008)Google Scholar
  194. 194.
    I. Gutman, A. Kaplarević, A. Nikolić, An auxiliary function in the theory of total π-electron energy. Kragujevac J. Sci. 23, 75–88 (2001)Google Scholar
  195. 195.
    I. Gutman, D. Kiani, M. Mirzakhah, On incidence energy of graphs. MATCH Commun. Math. Comput. Chem. 62, 573–580 (2009)MathSciNetzbMATHGoogle Scholar
  196. 196.
    I. Gutman, D. Kiani, M. Mirzakhah, B. Zhou, On incidence energy of a graph. Lin. Algebra Appl. 431, 1223–1233 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  197. 197.
    I. Gutman, A. Klobučar, S. Majstorović, C. Adiga, Biregular graphs whose energy exceeds the number of vertices. MATCH Commun. Math. Comput. Chem. 62, 499–508 (2009)MathSciNetzbMATHGoogle Scholar
  198. 198.
    I. Gutman, J.H. Koolen, V. Moulton, M. Parac, T. Soldatović, D. Vidović, Estimating and approximating the total π-electron energy of benzenoid hydrocarbons. Z. Naturforsch. 55a, 507–512 (2000)Google Scholar
  199. 199.
    I. Gutman, X. Li, Y. Shi, J. Zhang, Hypoenergetic trees. MATCH Commun. Math. Comput. Chem. 60, 415–426 (2008)MathSciNetzbMATHGoogle Scholar
  200. 200.
    I. Gutman, X. Li, J. Zhang, in Graph Energy, ed. by M. Dehmer, F. Emmert–Streib. Analysis of Complex Networks. From Biology to Linguistics (Wiley–VCH, Weinheim, 2009), pp. 145–174Google Scholar
  201. 201.
    I. Gutman, S. Marković, Topological properties of benzenoid systems. XLVIIIa. An empirical study of two contradictory formulas for total π-electron energy. MATCH Commun. Math. Comput. Chem. 25, 141–149 (1990)Google Scholar
  202. 202.
    I. Gutman, S. Marković, G.G. Hall, Revisiting a simple regularity for benzenoid hydrocarbons: Total π-electron energy versus the number of Kekulé structures. Chem. Phys. Lett. 234, 21–24 (1995)CrossRefGoogle Scholar
  203. 203.
    I. Gutman, S. Marković, M. Marinković, Investigation of the Cioslowski formula. MATCH Commun. Math. Comput. Chem. 22, 277–284 (1987)Google Scholar
  204. 204.
    I. Gutman, S. Marković, A.V. Teodorović, Ž. Bugarčić, Isomer–undistinguishing approximate formulas for the total π-electron energy of benzenoid hydrocarbons. J. Serb. Chem. Soc. 51, 145–149 (1986)Google Scholar
  205. 205.
    I. Gutman, S. Marković, A. Vesović, E. Estrada, Approximating total π-electron energy in terms of spectral moments. A quantitative approach. J. Serb. Chem. Soc. 63, 639–646 (1998)Google Scholar
  206. 206.
    I. Gutman, S. Marković, D. Vukićević, A. Stajković, The dependence of total π-electron energy of large benzenoid hydrocarbons on the number of Kekulé structures is non-linear. J. Serb. Chem. Soc. 60, 93–98 (1995)Google Scholar
  207. 207.
    I. Gutman, M. Mateljević, Note on the Coulson integral formula. J. Math. Chem. 39, 259–266 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  208. 208.
    I. Gutman, M. Milun, N. Trinajstić, Comment on the paper: “Properties of the latent roots of a matrix. Estimation of π-electron energies” ed. by B.J. McClelland. J. Chem. Phys. 59, 2772–2774 (1973)Google Scholar
  209. 209.
    I. Gutman, M. Milun, N. Trinajstić, Graph theory and molecular orbitals. 19. Nonparametric resonance energies of arbitrary conjugated systems. J. Am. Chem. Soc. 99, 1692–1704 (1977)Google Scholar
  210. 210.
    I. Gutman, L. Nedeljković, A.V. Teodorović, Topological formulas for total π-electron energy of benzenoid hydrocarbons – a comparative study. Bull. Soc. Chim. Beograd 48, 495–500 (1983)Google Scholar
  211. 211.
    I. Gutman, A. Nikolić, Ž. Tomović, A concealed property of total π-electron energy. Chem. Phys. Lett. 349, 95–98 (2001)CrossRefGoogle Scholar
  212. 212.
    I. Gutman, L. Pavlović, The energy of some graphs with large number of edges. Bull. Acad. Serbe Sci. Arts. (Cl. Math. Natur.) 118, 35–50 (1999)Google Scholar
  213. 213.
    I. Gutman, S. Petrović, On total π-electron energy of benzenoid hydrocarbons. Chem. Phys. Lett. 97, 292–294 (1983)CrossRefGoogle Scholar
  214. 214.
    I. Gutman, P. Petković, P.V. Khadikar, Bounds for the total π-electron energy of phenylenes. Rev. Roum. Chim. 41, 637–643 (1996)Google Scholar
  215. 215.
    I. Gutman, O.E. Polansky, Cyclic conjugation and the Hückel molecular orbital model. Theor. Chim. Acta 60, 203–226 (1981)Google Scholar
  216. 216.
    I. Gutman, O.E. Polansky, Mathematical Concepts in Organic Chemistry (Springer, Berlin, 1986)zbMATHCrossRefGoogle Scholar
  217. 217.
    I. Gutman, S. Radenković, Extending and modifying the Hall rule. Chem. Phys. Lett. 423, 382–385 (2006)CrossRefGoogle Scholar
  218. 218.
    I. Gutman, S. Radenković, Hypoenergetic molecular graphs. Indian J. Chem. 46A, 1733–1736 (2007)Google Scholar
  219. 219.
    I. Gutman, S. Radenković, N. Li, S. Li, Extremal energy of trees. MATCH Commun. Math. Comput. Chem. 59, 315–320 (2008)MathSciNetzbMATHGoogle Scholar
  220. 220.
    I. Gutman, M. Rašković, Monte Carlo approach to total π-electron energy of conjugated hydrocarbons. Z. Naturforsch. 40a, 1059–1061 (1985)Google Scholar
  221. 221.
    I. Gutman, M. Robbiano, E. Andrade–Martins, D.M. Cardoso, L. Medina, O. Rojo, Energy of line graphs. Lin. Algebra Appl. 433, 1312–1323 (2010)Google Scholar
  222. 222.
    I. Gutman, B. Ruščić, N. Trinajstić, C.F. Wilcox, Graph theory and molecular orbitals. XII. Acyclic polyenes. J. Chem. Phys. 62, 3399–3405 (1975)Google Scholar
  223. 223.
    I. Gutman, J.Y. Shao, The energy change of weighted graphs. Lin. Algebra Appl. 435, 2425–2431 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  224. 224.
    I. Gutman, T. Soldatović, Novel approximate formulas for the total π-electron energy of benzenoid hydrocarbons. Bull. Chem. Technol. Maced. 19, 17–20 (2000)Google Scholar
  225. 225.
    I. Gutman, T. Soldatović, (n, m)-Type approximations for total π-electron energy of benzenoid hydrocarbons. MATCH Commun. Math. Comput. Chem. 44, 169–182 (2001)Google Scholar
  226. 226.
    I. Gutman, T. Soldatović, On a class of approximate formulas for total π-electron energy of benzenoid hydrocarbons. J. Serb. Chem. Soc. 66, 101–106 (2001)Google Scholar
  227. 227.
    I. Gutman, T. Soldatović, A. Graovac, S. Vuković, Approximating the total π-electron energy by means of spectral moments. Chem. Phys. Lett. 334, 168–172 (2001)CrossRefGoogle Scholar
  228. 228.
    I. Gutman, T. Soldatović, M. Petković, A new upper bound and approximation for total π-electron energy. Kragujevac J. Sci. 23, 89–98 (2001)Google Scholar
  229. 229.
    I. Gutman, T. Soldatović, D. Vidović, The energy of a graph and its size dependence. A Monte Carlo approach. Chem. Phys. Lett. 297, 428–432 (1998)Google Scholar
  230. 230.
    I. Gutman, A. Stajković, S. Marković, P. Petković, Dependence of total π-electron energy of phenylenes on Kekulé structure count. J. Serb. Chem. Soc. 59, 367–373 (1994)Google Scholar
  231. 231.
    I. Gutman, S. Stanković, J. Durdević, B. Furtula, On the cycle–dependence of topological resonance energy. J. Chem. Inf. Model. 47, 776–781 (2007)CrossRefGoogle Scholar
  232. 232.
    I. Gutman, D. Stevanović, S. Radenković, S. Milosavljević, N. Cmiljanović, Dependence of total π-electron energy on large number of non-bonding molecular orbitals. J. Serb. Chem. Soc. 69, 777–782 (2004)CrossRefGoogle Scholar
  233. 233.
    I. Gutman, A.V. Teodorović, Ž. Bugarčić, On some topological formulas for total π-electron energy of benzenoid molecules. Bull. Soc. Chim. Beograd 49, 521–525 (1984)Google Scholar
  234. 234.
    I. Gutman, A.V. Teodorović, L. Nedeljković, Topological properties of benzenoid systems. Bounds and approximate formulae for total π-electron energy. Theor. Chim. Acta 65, 23–31 (1984)Google Scholar
  235. 235.
    I. Gutman, Ž. Tomović, Total π-electron energy of phenylenes: Bounds and approximate expressions. Monatsh. Chem. 132, 1023–1029 (2001)CrossRefGoogle Scholar
  236. 236.
    I. Gutman, N. Trinajstić, Graph theory and molecular orbitals. Total π-electron energy of alternant hydrocarbons. Chem. Phys. Lett. 17, 535–538 (1972)Google Scholar
  237. 237.
    I. Gutman, N. Trinajstić, Graph theory and molecular orbitals. The loop rule. Chem. Phys. Lett. 20, 257–260 (1973)Google Scholar
  238. 238.
    I. Gutman, N. Trinajstić, Graph theory and molecular orbitals. Topics Curr. Chem. 42, 49–93 (1973)Google Scholar
  239. 239.
    I. Gutman, L. Türker, Approximating the total π-electron energy of benzenoid hydrocarbons: Some new estimates of (n, m)-type. Indian J. Chem. 32A, 833–836 (1993)Google Scholar
  240. 240.
    I. Gutman, L. Türker, J.R. Dias, Another upper bound for total π-electron energy of alternant hydrocarbons. MATCH Commun. Math. Comput. Chem. 19, 147–161 (1986)Google Scholar
  241. 241.
    I. Gutman, D. Utvić, A.K. Mukherjee, A class of topological formulas for total π-electron energy. J. Serb. Chem. Soc. 56, 59–63 (1991)Google Scholar
  242. 242.
    I. Gutman, D. Vidović, Quest for molecular graphs with maximal energy: A computer experiment. J. Chem. Inf. Comput. Sci. 41, 1002–1005 (2001)CrossRefGoogle Scholar
  243. 243.
    I. Gutman, D. Vidović, Conjugated molecules with maximal total π-electron energy. Bull. Acad. Serbe Sci. Arts (Cl. Math. Natur.) 124, 1–7 (2003)Google Scholar
  244. 244.
    I. Gutman, D. Vidović, N. Cmiljanović, S. Milosavljević, S. Radenković, Graph energy – A useful molecular structure-descriptor. Indian J. Chem. 42A, 1309–1311 (2003)Google Scholar
  245. 245.
    I. Gutman, D. Vidović, H. Hosoya, The relation between the eigenvalue sum and the topological index Z revisited. Bull. Chem. Soc. Jpn. 75, 1723–1727 (2002)CrossRefGoogle Scholar
  246. 246.
    I. Gutman, D. Vidović, T. Soldatović, Modeling the dependence of the π-electron energy on the size of conjugated molecules. A Monte Carlo approach. ACH – Models Chem. 136, 599–608 (1999)Google Scholar
  247. 247.
    I. Gutman, S. Zare Firoozabadi, J.A. de la Penña, J. Rada, On the energy of regular graphs. MATCH Commun. Math. Comput. Chem. 57, 435–442 (2007)MathSciNetzbMATHGoogle Scholar
  248. 248.
    I. Gutman, F. Zhang, On the quasiordering of bipartite graphs. Publ. Inst. Math. (Belgrade) 40, 11–15 (1986)MathSciNetGoogle Scholar
  249. 249.
    I. Gutman, F. Zhang, On the ordering of graphs with respect to their matching numbers. Discr. Appl. Math. 15, 25–33 (1986)MathSciNetzbMATHCrossRefGoogle Scholar
  250. 250.
    I. Gutman, B. Zhou, Laplacian energy of a graph. Lin. Algebra Appl. 414, 29–37 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  251. 251.
    I. Gutman, B. Zhou, B. Furtula, The Laplacian-energy like invariant is an energy like invariant. MATCH Commun. Math. Comput. Chem. 64, 85–96 (2010)MathSciNetGoogle Scholar
  252. 252.
    W.H. Haemers, Strongly regular graphs with maximal energy. Lin. Algebra Appl. 429, 2719–2723 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  253. 253.
    W.H. Haemers, Q. Xiang, Strongly regular graphs with parameters (4m 4, 2m 4 + m 2, m 4 + m 2, m 4 + m 2) exist for all m > 1. Eur. J. Comb. 31, 1553–1559 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  254. 254.
    G.G. Hall, The bond orders of alternant hydrocarbon molecules. Proc. Roy. Soc. A 229, 251–259 (1955)Google Scholar
  255. 255.
    G.G. Hall, A graphical model of a class of molecules. Int. J. Math. Educ. Sci. Technol. 4, 233–240 (1973)CrossRefGoogle Scholar
  256. 256.
    M. Hall, Combinatorial Theory (Wiley, New York, 1986)zbMATHGoogle Scholar
  257. 257.
    C.X. He, B.F. Wu, Z.S. Yu, On the energy of trees with given domination number. MATCH Commun. Math. Comput. Chem. 64, 169–180 (2010)MathSciNetGoogle Scholar
  258. 258.
    C. Heuberger, H. Prodinger, S. Wagner, Positional number systems with digits forming an arithmetic progression. Monatsh. Math. 155, 349–375 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  259. 259.
    C. Heuberger, S. Wagner, Maximizing the number of independent subsets over trees with bounded degree. J. Graph Theor. 58, 49–68 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  260. 260.
    C. Heuberger, S. Wagner, Chemical trees minimizing energy and Hosoya index. J. Math. Chem. 46, 214–230 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  261. 261.
    C. Heuberger, S. Wagner, On a class of extremal trees for various indices. MATCH Commun. Math. Comput. Chem. 62, 437–464 (2009)MathSciNetzbMATHGoogle Scholar
  262. 262.
    M. Hofmeister, Spectral radius and degree sequence. Math. Nachr. 139, 37–44 (1988)MathSciNetzbMATHCrossRefGoogle Scholar
  263. 263.
    V.E. Hoggatt, M. Bicknell, Roots of Fibonacci polynomials. Fibonacci Quart. 11, 271–274 (1973)MathSciNetzbMATHGoogle Scholar
  264. 264.
    Y. Hong, X. Zhang, Sharp upper and lower bounds for largest eigenvalue of the Laplacian matrices of trees. Discr. Math 296, 187–197 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  265. 265.
    R. Horn, C. Johnson, Matrix Analysis (Cambridge University Press, Cambridge, 1989)zbMATHGoogle Scholar
  266. 266.
    Y. Hou, Unicyclic graphs with minimal energy. J. Math. Chem. 29, 163–168 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  267. 267.
    Y. Hou, Bicyclic graphs with minimum energy. Lin. Multilin. Algebra 49, 347–354 (2001)zbMATHCrossRefGoogle Scholar
  268. 268.
    Y. Hou, On trees with the least energy and a given size of matching. J. Syst. Sci. Math. Sci. 23, 491–494 (2003) [in Chinese]Google Scholar
  269. 269.
    Y. Hou, I. Gutman, Hyperenergetic line graphs. MATCH Commun. Math. Comput. Chem. 43, 29–39 (2001)MathSciNetzbMATHGoogle Scholar
  270. 270.
    Y. Hou, I. Gutman, C.W. Woo, Unicyclic graphs with maximal energy. Lin. Algebra Appl. 356, 27–36 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  271. 271.
    Y. Hou, Z. Teng, C. Woo, On the spectral radius, k-degree and the upper bound of energy in a graph. MATCH Commun. Math. Comput. Chem. 57, 341–350 (2007)MathSciNetzbMATHGoogle Scholar
  272. 272.
    X. Hu, H. Liu, New upper bounds for the Hückel energy of graphs. MATCH Commun. Math. Comput. Chem. 66, 863–878 (2011)MathSciNetGoogle Scholar
  273. 273.
    H. Hua, On minimal energy of unicyclic graphs with prescribed girth and pendent vertices. MATCH Commun. Math. Comput. Chem. 57, 351–361 (2007)MathSciNetzbMATHGoogle Scholar
  274. 274.
    H. Hua, Bipartite unicyclic graphs with large energy. MATCH Commun. Math. Comput. Chem. 58, 57–83 (2007)MathSciNetzbMATHGoogle Scholar
  275. 275.
    H. Hua, M. Wang, Unicyclic graphs with given number of pendent vertices and minimal energy. Lin. Algebra Appl. 426, 478–489 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  276. 276.
    X. Hui, H. Deng, Solutions of some unsolved problems on hypoenergetic unicyclic, bicyclic and tricyclic graphs. MATCH Commun. Math. Comput. Chem. 64, 231–238 (2010)MathSciNetGoogle Scholar
  277. 277.
    B. Huo, S. Ji, X. Li, Note on unicyclic graphs with given number of pendent vertices and minimal energy. Lin. Algebra Appl. 433, 1381–1387 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  278. 278.
    B. Huo, S. Ji, X. Li, Solutions to unsolved problems on the minimal energies of two classes of graphs. MATCH Commun. Math. Comput. Chem. 66, 943–958 (2011)MathSciNetGoogle Scholar
  279. 279.
    B. Huo, S. Ji, X. Li, Y. Shi, Complete solution to a conjecture on the fourth maximal energy tree. MATCH Commun. Math. Comput. Chem. 66, 903–912 (2011)MathSciNetGoogle Scholar
  280. 280.
    B. Huo, S. Ji, X. Li, Y. Shi, Solution to a conjecture on the maximal energy of bipartite bicyclic graphs. Lin. Algebra Appl. 435, 804–810 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  281. 281.
    B. Huo, X. Li, Y. Shi, Complete solution of a problem on the maximal energy of unicyclic bipartite graphs. Lin. Algebra Appl. 434, 1370–1377 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  282. 282.
    B. Huo, X. Li, Y. Shi, Complete solution to a conjecture on the maximal energy of unicyclic graphs. Eur. J. Comb. 32, 662–673 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  283. 283.
    B. Huo, X. Li, Y. Shi, L. Wang, Determining the conjugated trees with the third – through the six-minimal energies. MATCH Commun. Math. Comput. Chem. 65, 521–532 (2011)MathSciNetGoogle Scholar
  284. 284.
    A. Ilić, The energy of unitary Cayley graph. Lin. Algebra Appl. 431, 1881–1889 (2009)zbMATHCrossRefGoogle Scholar
  285. 285.
    A. Ilić, Distance spectra and distance energy of integral circulant graphs. Lin. Algebra Appl. 433, 1005–1014 (2010)zbMATHCrossRefGoogle Scholar
  286. 286.
    A. Ilić, M. Bašić, New results on the energy of integral circulant graphs. Appl. Math. Comput. 218, 3470–3482 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  287. 287.
    A. Ilić, M. Bašić, I. Gutman, Triply equienergetic graphs. MATCH Commun. Math. Comput. Chem. 64, 189–200 (2010)MathSciNetGoogle Scholar
  288. 288.
    A. Ilić, D-. Krtinić, M. Ilić, On Laplacian like energy of trees. MATCH Commun. Math. Comput. Chem. 64, 111–122 (2010)Google Scholar
  289. 289.
    G. Indulal, Sharp bounds on the distance spectral radius and the distance energy of graphs. Lin. Algebra Appl. 430, 106–113 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  290. 290.
    G. Indulal, I. Gutman, D-Equienergetic self-complementary graphs. Kragujevac J. Math. 32, 123–131 (2009)MathSciNetzbMATHGoogle Scholar
  291. 291.
    G. Indulal, I. Gutman, A. Vijayakumar, On distance energy of graphs. MATCH Commun. Math. Comput. Chem. 60, 461–472 (2008)MathSciNetzbMATHGoogle Scholar
  292. 292.
    G. Indulal, A. Vijayakumar, On a pair of equienergetic graphs. MATCH Commun. Math. Comput. Chem. 55, 83–90 (2006)MathSciNetzbMATHGoogle Scholar
  293. 293.
    G. Indulal, A. Vijayakumar, Energies of some non-regular graphs. J. Math. Chem. 42, 377–386 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  294. 294.
    G. Indulal, A. Vijayakumar, Classes of Türker equivalent graphs. Graph Theor. Notes New York 53, 30–36 (2007)MathSciNetGoogle Scholar
  295. 295.
    G. Indulal, A. Vijayakumar, A note on energy of some graphs. MATCH Commun. Math. Comput. Chem. 59, 269–274 (2008)MathSciNetGoogle Scholar
  296. 296.
    G. Indulal, A. Vijayakumar, Equienergetic self-complementary graphs. Czech. Math. J. 58, 911–919 (2008)MathSciNetzbMATHGoogle Scholar
  297. 297.
    Y. Jiang, A. Tang, R. Hoffmann, Evaluation of moments and their application to Hückel molecular orbital theory. Theor. Chim. Acta 65, 255–265 (1984)CrossRefGoogle Scholar
  298. 298.
    Y. Jiang, H. Zhu, H. Zhang, I. Gutman, Moment expansion of Hückel molecular energies. Chem. Phys. Lett. 159, 159–164 (1989)CrossRefGoogle Scholar
  299. 299.
    M.R. Jooyandeh, D. Kiani, M. Mirzakhah, Incidence energy of a graph. MATCH Commun. Math. Comput. Chem. 62, 561–572 (2009)MathSciNetzbMATHGoogle Scholar
  300. 300.
    I. Jovanović, Z. Stanić, Spectral distances of graphs. Lin. Algebra Appl. 436, 1425–1435 (2012)zbMATHCrossRefGoogle Scholar
  301. 301.
    H. Kharaghani, B. Tayfeh–Rezaie, On the energy of (0, 1)-matrices. Lin. Algebra Appl. 429, 2046–2051 (2008)Google Scholar
  302. 302.
    D. Kiani, M.M.H. Aghaei, Y. Meemark, B. Suntornpoch, Energy of unitary Cayley graphs and gcd-graphs. Lin. Algebra Appl. 435, 1336–1343 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  303. 303.
    D.J. Klein, V.R. Rosenfeld, Phased graphs and graph energies. J. Math. Chem. 49, 1238–1244 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  304. 304.
    D.J. Klein, V.R. Rosenfeld, Phased cycles. J. Math. Chem. 49, 1245–1255 (2011)MathSciNetzbMATHGoogle Scholar
  305. 305.
    J.H. Koolen, V. Moulton, Maximal energy graphs. Adv. Appl. Math. 26, 47–52 (2001)MathSciNetzbMATHGoogle Scholar
  306. 306.
    J.H. Koolen, V. Moulton, Maximal energy bipartite graphs. Graphs Combin. 19, 131–135 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  307. 307.
    J.H. Koolen, V. Moulton, I. Gutman, Improving the McClelland inequality for total π-electron energy. Chem. Phys. Lett. 320, 213–216 (2000)CrossRefGoogle Scholar
  308. 308.
    J.H. Koolen, V. Moulton, I. Gutman, D. Vidović, More hyperenergetic molecular graphs. J. Serb. Chem. Soc. 65, 571–575 (2000)Google Scholar
  309. 309.
    S. Lang, Algebra (Addison–Wesley, Reading, 1993)Google Scholar
  310. 310.
    B. Lass, Matching polynomials and duality. Combinatorica 24, 427–440 (2004)MathSciNetzbMATHGoogle Scholar
  311. 311.
    C.K. Li, W. So, Graphs equienergetic with edge-deleted subgraphs. Lin. Multilin. Algebra 57, 683–693 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  312. 312.
    F. Li, B. Zhou, Minimal energy of bipartite unicyclic graphs of a given biaprtition. MATCH Commun. Math. Comput. Chem. 54, 379–388 (2005)MathSciNetzbMATHGoogle Scholar
  313. 313.
    F. Li, B. Zhou, Minimal energy of unicyclic graphs of a given diameter. J. Math. Chem. 43, 476–484 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  314. 314.
    H. Li, On minimal energy ordering of acyclic conjugated molecules. J. Math. Chem. 25, 145–169 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  315. 315.
    J. Li, X. Li, Note on bipartite unicyclic graphs of a given bipartition with minimal energy. MATCH Commun. Math. Comput. Chem. 64, 61–64 (2010)MathSciNetGoogle Scholar
  316. 316.
    J. Li, X. Li, Y. Shi, On the maximal energy tree with two maximum degree vertices. Lin. Algebra Appl. 435, 2272–2284 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  317. 317.
    J. Li, X. Li, On the maximal energy trees with one maximum and one second maximum degree vertex. MATCH Commun. Math. Comput. Chem. 67, 525–539 (2012)MathSciNetGoogle Scholar
  318. 318.
    J. Li, X. Wang, Lower bound on the sum of positive eigenvalues of a graph. Acta Appl. Math. 14, 443–446 (1998)zbMATHCrossRefGoogle Scholar
  319. 319.
    N. Li, S. Li, On the extremal energy of trees. MATCH Commun. Math. Comput. Chem. 59, 291–314 (2008)MathSciNetzbMATHGoogle Scholar
  320. 320.
    R. Li, The spectral moments and energy of graphs. Appl. Math. Sci. 3, 2765–2773 (2009)MathSciNetzbMATHGoogle Scholar
  321. 321.
    R. Li, Energy and some Hamiltonian properties of graphs. Appl. Math. Sci. 3, 2775–2780 (2009)MathSciNetzbMATHGoogle Scholar
  322. 322.
    R. Li, Some lower bounds for Laplacian energy of graphs. Int. J. Contemp. Math. Sci. 4, 219–233 (2009)MathSciNetzbMATHGoogle Scholar
  323. 323.
    R. Li, On α-incidence energy and α-distance energy of a graph. Ars Combin. in pressGoogle Scholar
  324. 324.
    S. Li, N. Li, On minimal energies of acyclic conjugated molecules. MATCH Commun. Math. Comput. Chem. 61, 341–349 (2009)MathSciNetGoogle Scholar
  325. 325.
    S. Li, X. Li, On tetracyclic graphs with minimal energy. MATCH Commun. Math. Comput. Chem. 60, 395–414 (2008)MathSciNetzbMATHGoogle Scholar
  326. 326.
    S. Li, X. Li, On tricyclic graphs of a given diameter with minimal energy. Lin. Algebra Appl. 430, 370–385 (2009)zbMATHCrossRefGoogle Scholar
  327. 327.
    S. Li, X. Li, The fourth maximal energy of acyclic graphs. MATCH Commun. Math. Comput. Chem. 61, 383–394 (2009)MathSciNetzbMATHGoogle Scholar
  328. 328.
    S. Li, X. Li, H. Ma, I. Gutman, On triregular graphs whose energy exceeds the number of vertices. MATCH Commun. Math. Comput. Chem. 64, 201–216 (2010)MathSciNetGoogle Scholar
  329. 329.
    S. Li, X. Li, Z. Zhu, On tricyclic graphs with minimal energy. MATCH Commun. Math. Comput. Chem. 59, 397–419 (2008)MathSciNetzbMATHGoogle Scholar
  330. 330.
    S. Li, X. Li, Z. Zhu, On minimal energy and Hosoya index of unicyclic graphs. MATCH Commun. Math. Comput. Chem. 61, 325–339 (2009)MathSciNetzbMATHGoogle Scholar
  331. 331.
    X. Li, Y. Li, Note on conjugated unicyclic graphs with minimal energy. MATCH Commun. Math. Comput. Chem. 64, 141–144 (2010)MathSciNetGoogle Scholar
  332. 332.
    X. Li, Y. Li, Y. Shi, Note on the energy of regular graphs. Lin. Algebra Appl. 432, 1144–1146 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  333. 333.
    X. Li, H. Lian, Conjugated chemical trees with extremal energy. MATCH Commun. Math. Comput. Chem. 66, 891–902 (2011)MathSciNetGoogle Scholar
  334. 334.
    X. Li, J. Liu, Note for Nikiforov’s two conjectures on the energy of trees, arXiv:0906.0827Google Scholar
  335. 335.
    X. Li, H. Ma, All connected graphs with maximum degree at most 3 whose energies are equal to the number of vertices. MATCH Commun. Math. Comput. Chem. 64, 7–24 (2010)MathSciNetGoogle Scholar
  336. 336.
    X. Li, H. Ma, Hypoenergetic and strongly hypoenergetic k-cyclic graphs. MATCH Commun. Math. Comput. Chem. 64, 41–60 (2010)MathSciNetGoogle Scholar
  337. 337.
    X. Li, H. Ma, Hypoenergetic and strongly hypoenergetic trees, arXiv:0905.3944.Google Scholar
  338. 338.
    X. Li, H. Ma, All hypoenergetic graphs with maximum degree at most 3. Lin. Algebra Appl. 431, 2127–2133 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  339. 339.
    X. Li, X. Yao, J. Zhang, I. Gutman, Maximum energy trees with two maximum degree vertices. J. Math. Chem. 45, 962–973 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  340. 340.
    X. Li, J. Zhang, On bicyclic graphs with maximal energy. Lin. Algebra Appl. 427, 87–98 (2007)zbMATHCrossRefGoogle Scholar
  341. 341.
    X. Li, J. Zhang, L. Wang, On bipartite graphs with minimal energy. Discr. Appl. Math. 157, 869–873 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  342. 342.
    X. Li, J. Zhang, B. Zhou, On unicyclic conjugated molecules with minimal energies. J. Math. Chem. 42, 729–740 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  343. 343.
    X. Lin, X. Guo, On the minimal energy of trees with a given number of vertices of degree two. MATCH Commun. Math. Comput. Chem. 62, 473–480 (2009)MathSciNetzbMATHGoogle Scholar
  344. 344.
    W. Lin, X. Guo, H. Li, On the extremal energies of trees with a given maximum degree. MATCH Commun. Math. Comput. Chem. 54, 363–378 (2005)MathSciNetzbMATHGoogle Scholar
  345. 345.
    W. Lin, W. Yan, Laplacian coefficients of trees with a given bipartition. Lin. Algebra Appl. 435, 152–162 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  346. 346.
    B. Liu, Y. Huang, Z. You, A survey on the Laplacian-energy-like invariant. MATCH Commun. Math. Comput. Chem. 66, 713–730 (2011)MathSciNetGoogle Scholar
  347. 347.
    H. Liu, M. Lu, Sharp bounds on the spectral radius and the energy of graphs. MATCH Commun. Math. Comput. Chem. 59, 279–290 (2008)MathSciNetzbMATHGoogle Scholar
  348. 348.
    H. Liu, M. Lu, F. Tian, Some upper bounds for the energy of graphs. J. Math. Chem. 41, 45–57 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  349. 349.
    J. Liu, B. Liu, Note on a pair of equienergetic graphs. MATCH Commun. Math. Comput. Chem. 59, 275–278 (2008)MathSciNetzbMATHGoogle Scholar
  350. 350.
    J. Liu, B. Liu, A Laplacian–energy like invariant of a graph. MATCH Commun. Math. Comput. Chem. 59, 355–372 (2008)MathSciNetzbMATHGoogle Scholar
  351. 351.
    J. Liu, B. Liu, On relation between energy and Laplacian energy. MATCH Commun. Math. Comput. Chem. 61, 403–406 (2009)MathSciNetzbMATHGoogle Scholar
  352. 352.
    J. Liu, B. Liu, On a conjecture about the hypoenergetic trees. Appl. Math. Lett. 23, 484–486 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  353. 353.
    J. Liu, B. Liu, E-L equienergetic graphs. MATCH Commun. Math. Comput. Chem. 66, 971–976 (2011)MathSciNetGoogle Scholar
  354. 354.
    J. Liu, B. Liu, S. Radenković, I. Gutman, Minimal LEL–equienergetic graphs. MATCH Commun. Math. Comput. Chem. 61, 471–478 (2009)MathSciNetzbMATHGoogle Scholar
  355. 355.
    M. Liu, A note on D-equienergetic graphs. MATCH Commun. Math. Comput. Chem. 64, 125–140 (2010)Google Scholar
  356. 356.
    M. Liu, B. Liu, A note on the LEL-equienergetic graphs. Ars Comb. in pressGoogle Scholar
  357. 357.
    Y. Liu, Some results on energy of unicyclic graphs with n vertices. J. Math. Chem. 47, 1–10 (2010)MathSciNetCrossRefGoogle Scholar
  358. 358.
    Z. Liu, B. Zhou, Minimal energies of bipartite bicyclic graphs. MATCH Commun. Math. Comput. Chem. 59, 381–396 (2008)MathSciNetzbMATHGoogle Scholar
  359. 359.
    W. López, J. Rada, Equienergetic digraphs. Indian J. Pure Appl. Math. 36, 361–372 (2007)zbMATHGoogle Scholar
  360. 360.
    L. Lovász, J. Pelikán, On the eigenvalues of trees. Period. Math. Hungar. 3, 175–182 (1973)MathSciNetzbMATHCrossRefGoogle Scholar
  361. 361.
    S. Majstorović, I. Gutman, A. Klobučar, Tricyclic biregular graphs whose energy exceeds the number of vertices. Math. Commun. 15, 213–222 (2010)MathSciNetzbMATHGoogle Scholar
  362. 362.
    S. Majstorović, A. Klobučar, I. Gutman, Triregular graphs whose energy exceeds the number of vertices. MATCH Commun. Math. Comput. Chem. 62, 509–524 (2009)MathSciNetzbMATHGoogle Scholar
  363. 363.
    S. Majstorović, A. Klobučar, I. Gutman, in Selected Topics from the Theory of Graph Energy: Hypoenergetic Graphs, ed. by D. Cvetković, I. Gutman. Applications of Graph Spectra (Mathematical Institute, Belgrade, 2009), pp. 65–105Google Scholar
  364. 364.
    M. Marcus, H. Minc, A Survey of Matrix Theory and Matrix Inequalities (Dover, New York, 1992)Google Scholar
  365. 365.
    S. Marković, Approximating total π-electron energy of phenylenes in terms of spectral moments. Indian J. Chem. 42A, 1304–1308 (2003)Google Scholar
  366. 366.
    M. Mateljević, V. Božin, I. Gutman, Energy of a polynomial and the Coulson integral formula. J. Math. Chem. 48, 1062–1068 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  367. 367.
    M. Mateljević, I. Gutman, Note on the Coulson and Coulson–Jacobs integral formulas. MATCH Commun. Math. Comput. Chem. 59, 257–268 (2008)MathSciNetzbMATHGoogle Scholar
  368. 368.
    B.J. McClelland, Properties of the latent roots of a matrix: The estimation of π-electron energies. J. Chem. Phys. 54, 640–643 (1971)CrossRefGoogle Scholar
  369. 369.
    M.L. Mehta, Random Matrices (Academic, New York, 1991)zbMATHGoogle Scholar
  370. 370.
    R. Merris, The distance spectrum of a tree. J. Graph Theor. 14, 365–369 (1990)MathSciNetzbMATHCrossRefGoogle Scholar
  371. 371.
    R. Merris, Laplacian matrices of graphs: A survey. Lin. Algebra Appl. 197–198, 143–176 (1994)MathSciNetCrossRefGoogle Scholar
  372. 372.
    R. Merris, An inequality for eigenvalues of symmetric matrices with applications to max–cuts and graph energy. Lin. Multilin Algebra 36, 225–229 (1994)MathSciNetzbMATHCrossRefGoogle Scholar
  373. 373.
    R. Merris, A survey of graph Laplacians. Lin. Multilin. Algebra 39, 19–31 (1995)MathSciNetzbMATHCrossRefGoogle Scholar
  374. 374.
    O. Miljković, B. Furtula, S. Radenković, I. Gutman, Equienergetic and almost–equienergetic trees. MATCH Commun. Math. Comput. Chem. 61, 451–461 (2009)MathSciNetzbMATHGoogle Scholar
  375. 375.
    B. Mohar, in The Laplacian Spectrum of Graphs, ed. by Y. Alavi, G. Chartrand, O.R. Oellermann, A.J. Schwenk. Graph Theory, Combinatorics, and Applications (Wiley, New York, 1991), pp. 871–898Google Scholar
  376. 376.
    D.A. Morales, Bounds for the total π-electron energy. Int. J. Quant. Chem. 88, 317–330 (2002)CrossRefGoogle Scholar
  377. 377.
    D.A. Morales, Systematic search for bounds for total π-electron energy. Int. J. Quant. Chem. 93, 20–31 (2003)CrossRefGoogle Scholar
  378. 378.
    D.A. Morales, The total π-electron energy as a problem of moments: Application of the Backus–Gilbert method. J. Math. Chem. 38, 389–397 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  379. 379.
    E. Munarini, Characteristic, admittance and matching polynomial of an antiregular graph. Appl. Anal. Discr. Math. 3, 157–176 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  380. 380.
    M. Muzychuk, Q. Xiang, Symmetric Bush-type Hadamard matrices of order 4m 4 exist for all odd m. Proc. Am. Math. Soc. 134, 2197–2204 (2006)Google Scholar
  381. 381.
    M.J. Nadjafi–Arani, Sharp bounds on the PI and vertex PI energy of graphs. MATCH Commun. Math. Chem. 65, 123–130 (2011)Google Scholar
  382. 382.
    V. Nikiforov, Walks and the spectral radius of graphs. Lin. Algebra Appl. 418, 257–268 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  383. 383.
    V. Nikiforov, The energy of graphs and matrices. J. Math. Anal. Appl. 326, 1472–1475 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  384. 384.
    V. Nikiforov, Graphs and matrices with maximal energy. J. Math. Anal. Appl. 327, 735–738 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  385. 385.
    V. Nikiforov, The energy of C 4-free graphs of bounded degree. Lin. Algebra Appl. 428, 2569–2573 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  386. 386.
    V. Nikiforov, On the sum of k largest singular values of graphs and matrices. Lin. Algebra Appl. 435, 2394–2401 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  387. 387.
    V. Nikiforov, Extremal norms of graphs and matrices. J. Math. Sci. 182, 164–174 (2012)zbMATHCrossRefGoogle Scholar
  388. 388.
    E.A. Nordhaus, B.M. Stewart, Triangles in an ordinary graph. Canad. J. Math. 15, 33–41 (1963)MathSciNetzbMATHCrossRefGoogle Scholar
  389. 389.
    J. Ou, On acyclic molecular graphs with maximal Hosoya index, energy, and short diameter. J. Math. Chem. 43, 328–337 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  390. 390.
    J. Ou, On ordering chemical trees by energy. MATCH Commun. Math. Comput. Chem. 64, 157–168 (2010)MathSciNetGoogle Scholar
  391. 391.
    J. Ou, Acyclic molecules with second maximal energy. Appl. Math. Lett. 23, 343–346 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  392. 392.
    I. Peña, J. Rada, Energy of digraphs. Lin. Multilin. Algebra 56, 565–579 (2008)zbMATHCrossRefGoogle Scholar
  393. 393.
    M. Perić, I. Gutman, J. Radić–Perić, The Hückel total π-electron energy puzzle. J. Serb. Chem. Soc. 71, 771–783 (2006)Google Scholar
  394. 394.
    S. Pirzada, I. Gutman, Energy of a graph is never the square root of an odd integer. Appl. Anal. Discr. Math. 2, 118–121 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  395. 395.
    J. Rada, Energy ordering of catacondensed hexagonal systems. Discr. Appl. Math. 145, 437–443 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  396. 396.
    J. Rada, The McClelland inequality for the energy of digraphs. Lin. Algebra Appl. 430, 800–804 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  397. 397.
    J. Rada, Lower bound for the energy of digraphs. Lin. Algebra Appl. 432, 2174–2169 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  398. 398.
    J. Rada, Bounds for the energy of normal digraphs. Lin. Multilin. Algebra 60, 323–332 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  399. 399.
    J. Rada, A. Tineo, Polygonal chains with minimal energy. Lin. Algebra Appl. 372, 333–344 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  400. 400.
    J. Rada, A. Tineo, Upper and lower bounds for the energy of bipartite graphs. J. Math. Anal. Appl. 289, 446–455 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  401. 401.
    S. Radenković, I. Gutman, Total π-electron energy and Laplacian energy: How far the analogy goes? J. Serb. Chem. Soc. 72, 1343–1350 (2007)CrossRefGoogle Scholar
  402. 402.
    H.S. Ramane, I. Gutman, D.S. Revankar, Distance equienergetic graphs. MATCH Commun. Math. Comput. Chem. 60, 473–484 (2008)MathSciNetzbMATHGoogle Scholar
  403. 403.
    H.S. Ramane, I. Gutman, H.B. Walikar, S.B. Halkarni, Another class of equienergetic graphs. Kragujevac J. Math. 26, 15–18 (2004)MathSciNetzbMATHGoogle Scholar
  404. 404.
    H.S. Ramane, I. Gutman, H.B. Walikar, S.B. Halkarni, Equienergetic complement graphs. Kragujevac J. Sci. 27, 67–74 (2005)Google Scholar
  405. 405.
    H.S. Ramane, D.S. Revankar, I. Gutman, S.B. Rao, B.D. Acharya, H.B. Walikar, Bounds for the distance energy of a graph. Kragujevac J. Math. 31, 59–68 (2008)MathSciNetzbMATHGoogle Scholar
  406. 406.
    H.S. Ramane, D.S. Revankar, I. Gutman, H.B. Walikar, Distance spectra and distance energies of iterated line graphs of regular graphs. Publ. Inst. Math. (Beograd) 85, 39–46 (2009)MathSciNetCrossRefGoogle Scholar
  407. 407.
    H.S. Ramane, H.B. Walikar, Construction of eqienergetic graphs. MATCH Commun. Math. Comput. Chem. 57, 203–210 (2007)MathSciNetzbMATHGoogle Scholar
  408. 408.
    H.S. Ramane, H.B. Walikar, I. Gutman, Equienergetic graphs. J. Comb. Math. Comb. Comput. 69, 165–173 (2009)MathSciNetzbMATHGoogle Scholar
  409. 409.
    H.S. Ramane, H.B. Walikar, S. Rao, B. Acharya, P. Hampiholi, S. Jog, I. Gutman, Equienergetic graphs. Kragujevac J. Math. 26, 5–13 (2004)MathSciNetzbMATHGoogle Scholar
  410. 410.
    H.S. Ramane, H.B. Walikar, S. Rao, B. Acharya, P. Hampiholi, S. Jog, I. Gutman, Spectra and energies of iterated line graphs of regular graphs. Appl. Math. Lett. 18, 679–682 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  411. 411.
    H.N. Ramaswamy, C.R. Veena, On the energy of unitary Cayley graphs. El. J. Combin. 16, #N24 (2009)Google Scholar
  412. 412.
    S.B. Rao, Energy of a graph, preprint, 2004Google Scholar
  413. 413.
    H. Ren, F. Zhang, Double hexagonal chains with minimal total π-electron energy. J. Math. Chem. 42, 1041–1056 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  414. 414.
    H. Ren, F. Zhang, Double hexagonal chains with maximal total energy. Int. J. Quant. Chem. 107, 1437–1445 (2007)CrossRefGoogle Scholar
  415. 415.
    H. Ren, F. Zhang, Fully–angular polyhex chains with minimal π-electron energy. J. Math. Anal. Appl. 326, 1244–1253 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  416. 416.
    M. Robbiano, E.A. Martins, I. Gutman, Extending a theorem by Fiedler and applications to graph energy. MATCH Commun. Math. Comput. Chem. 64, 145–156 (2010)MathSciNetGoogle Scholar
  417. 417.
    M. Robbiano, E. Andrade Martins, R. Jiménez, B. San Martín, Upper bounds on the Laplacian energy of some graphs. MATCH Commun. Math. Comput. Chem. 64, 97–110 (2010)MathSciNetGoogle Scholar
  418. 418.
    M. Robbiano, R. Jiménez, Applications of a theorem by Ky Fan in the theory of Laplacian energy of graphs. MATCH Commun. Math. Comput. Chem. 62, 537–552 (2009)MathSciNetzbMATHGoogle Scholar
  419. 419.
    M. Robbiano, R. Jiménez, Improved bounds for the Laplacian energy of Bethe trees. Lin. Algebra Appl. 432, 2222–2229 (2010)zbMATHCrossRefGoogle Scholar
  420. 420.
    M. Robbiano, R. Jiménez, L. Medina, The energy and an approximation to Estrada index of some trees. MATCH Commun. Math. Comput. Chem. 61, 369–382 (2009)MathSciNetzbMATHGoogle Scholar
  421. 421.
    O. Rojo, Line graph eigenvalues and line energy of caterpillars. Lin. Algebra Appl. 435, 2077–2086 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  422. 422.
    O. Rojo, R.D. Jiménez, Line graph of combinations of generalized Bethe trees: eigenvalues and energy. Lin. Algebra Appl. 435, 2402–2419 (2011)zbMATHCrossRefGoogle Scholar
  423. 423.
    O. Rojo, L. Medina, Constructing graphs with energy rE(G) where G is a bipartite graph. MATCH Commun. Math. Comput. Chem. 62, 465–472 (2009)MathSciNetzbMATHGoogle Scholar
  424. 424.
    O. Rojo, L. Medina, Construction of bipartite graphs having the same Randić energy. MATCH Commun. Math. Comput. Chem. 68, 805–814 (2012)Google Scholar
  425. 425.
    K. Ruedenberg, Theorem on the mobile bond orders of alternant conjugated systems. J. Chem. Phys. 29, 1232–1233 (1958)CrossRefGoogle Scholar
  426. 426.
    K. Ruedenberg, Quantum mechanics of mobile electrons in conjugated bond systems. III. Topological matrix as generatrix of bond orders. J. Chem. Phys. 34, 1884–1891 (1961)Google Scholar
  427. 427.
    E. Sampathkumar, On duplicate graphs. J. Indian Math. Soc. 37, 285–293 (1973)MathSciNetGoogle Scholar
  428. 428.
    J.W. Sander, T. Sander, The energy of integral circulant graphs with prime power order. Appl. Anal. Discr. Math. 5, 22–36 (2011)MathSciNetCrossRefGoogle Scholar
  429. 429.
    J.W. Sander, T. Sander, Integral circulant graphs of prime order with maximal energy. Lin. Algebra Appl. 435, 3212–3232 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  430. 430.
    L.J. Schaad, B.A. Hess, Hückel molecular orbital π resonance energies. The question of the σ structure. J. Am. Chem. Soc. 94, 3068–3074 (1972)CrossRefGoogle Scholar
  431. 431.
    T.G. Schmalz, T. Živković, D.J. Klein, Cluster expansion of the Hückel molecular orbital energy of acyclics: Application to pi resonance theory. Stud. Phys. Theor. Chem. 54, 173–190 (1988)Google Scholar
  432. 432.
    H.Y. Shan, J.Y. Shao, Graph energy change due to edge grafting operations and its application. MATCH Commun. Math. Comput. Chem. 64, 25–40 (2010)MathSciNetGoogle Scholar
  433. 433.
    H.Y. Shan, J.Y. Shao, F. Gong, Y. Liu, An edge grafting theorem on the energy of unicyclic and bipartite graphs. Lin. Algebra Appl. 433, 547–556 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  434. 434.
    H.Y. Shan, J.Y. Shao, S. Li, X. Li, On a conjecture on the tree with fourth greatest energy. MATCH Commun. Math. Comput. Chem. 64, 181–188 (2010)MathSciNetGoogle Scholar
  435. 435.
    J.Y. Shao, F. Gong, Z. Du, The extremal energies of weighted trees and forests with fixed total weight sum. MATCH Commun. Math. Comput. Chem. 66, 879–890 (2011)MathSciNetGoogle Scholar
  436. 436.
    J.Y. Shao, F. Gong, I. Gutman, New approaches for the real and complex integral formulas of the energy of a polynomial. MATCH Commun. Math. Comput. Chem. 66, 849–861 (2011)MathSciNetGoogle Scholar
  437. 437.
    X. Shen, Y. Hou, I. Gutman, X. Hui, Hyperenergetic graphs and cyclomatic number. Bull. Acad. Serbe Sci. Arts (Cl. Sci. Math. Natur.) 141, 1–8 (2010)Google Scholar
  438. 438.
    I. Shparlinski, On the energy of some circulant graphs. Lin. Algebra Appl. 414, 378–382 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  439. 439.
    J.H. Smith, in Some Properties of the Spectrum of a Graph, ed. by R. Guy, H. Hanani, N. Sauer, J. Schönheim. Combinatorial Structures and their Applications (Gordon and Breach, New York, 1970), pp. 403–406Google Scholar
  440. 440.
    W. So, Remarks on some graphs with large number of edges. MATCH Commun. Math. Comput. Chem. 61, 351–359 (2009)MathSciNetzbMATHGoogle Scholar
  441. 441.
    W. So, M. Robbiano, N.M.M. de Abreu, I. Gutman, Applications of a theorem by Ky Fan in the theory of graph energy. Lin. Algebra Appl. 432, 2163–2169 (2010)zbMATHCrossRefGoogle Scholar
  442. 442.
    I. Stanković, M. Milošević, D. Stevanović, Small and not so small equienergetic graphs. MATCH Commun. Math. Comput. Chem. 61, 443–450 (2009)MathSciNetzbMATHGoogle Scholar
  443. 443.
    N.F. Stepanov, V.M. Tatevskii, Approximate calculation of π-electron energy of aromatic condenased molecules by the Hückel MO LCAO method. Zh. Strukt. Khim. (in Russian) 2, 452–455 (1961)Google Scholar
  444. 444.
    D. Stevanović, Energy and NEPS of graphs. Lin. Multilin. Algebra 53, 67–74 (2005)zbMATHCrossRefGoogle Scholar
  445. 445.
    D. Stevanović, Laplacian–like energy of trees. MATCH Commun. Math. Comput. Chem. 61, 407–417 (2009)MathSciNetzbMATHGoogle Scholar
  446. 446.
    D. Stevanović, Large sets of noncospectral graphs with equal Laplacian energy. MATCH Commun. Math. Comput. Chem. 61, 463–470 (2009)MathSciNetzbMATHGoogle Scholar
  447. 447.
    D. Stevanović, Approximate energy of dendrimers. MATCH Commun. Math. Comput. Chem. 64, 65–73 (2010)MathSciNetGoogle Scholar
  448. 448.
    D. Stevanović, Oriented incidence energy and threshold graphs. Filomat 25, 1–8 (2011)CrossRefGoogle Scholar
  449. 449.
    D. Stevanović, N.M.M. de Abreu, M.A.A. de Freitas, C. Vinagre, R. Del-Vecchio, On the oriented incidence energy and decomposable graphs. Filomat 23, 243–249 (2009)CrossRefGoogle Scholar
  450. 450.
    D. Stevanović, A. Ilić, On the Laplacian coefficients of unicyclic graphs. Lin. Algebra Appl. 430, 2290–2300 (2009)zbMATHCrossRefGoogle Scholar
  451. 451.
    D. Stevanović, A. Ilić, C. Onişor, M.V. Diudea, LEL – A newly designed molecular descriptor. Acta Chim. Sloven. 56, 410–417 (2009)Google Scholar
  452. 452.
    D. Stevanović, G. Indulal, The distance spectrum and energy of the composition of regular graphs. Appl. Math. Lett. 22, 1136–1140 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  453. 453.
    D. Stevanović, I. Stanković, Remarks on hyperenergetic circulant graphs. Lin. Algebra Appl. 400, 345–348 (2005)zbMATHCrossRefGoogle Scholar
  454. 454.
    D. Stevanović, I. Stanković, M. Milošević, More on the relation between energy and Laplacian energy of graphs. MATCH Commun. Math. Comput. Chem. 61, 395–401 (2009)MathSciNetzbMATHGoogle Scholar
  455. 455.
    S. Strunkov, S. Sánchez, Energy spectral specifications for the graph reconstruction. Commun. Algebra 36, 309–314 (2008)zbMATHCrossRefGoogle Scholar
  456. 456.
    S. Tan, T. Song, On the Laplacian coefficients of trees with a perfect matching. Lin. Algebra Appl. 436, 595–617 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  457. 457.
    Z. Tang, Y. Hou, On incidence energy of trees. MATCH Commun. Math. Comput. Chem. 66, 977–984 (2011)MathSciNetGoogle Scholar
  458. 458.
    R.C. Thompson, Singular value inequalities for matrix sums and minors. Lin. Algebra Appl. 11, 251–269 (1975)zbMATHCrossRefGoogle Scholar
  459. 459.
    R.C. Thompson, Convex and concave functions of singular values of matrix sums. Pacific J. Math. 66, 285–290 (1976)MathSciNetzbMATHCrossRefGoogle Scholar
  460. 460.
    G.X. Tian, On the skew energy of orientations of hypercubes. Lin. Algebra Appl. 435, 2140–2149 (2011)zbMATHCrossRefGoogle Scholar
  461. 461.
    G.X. Tian, T.Z. Huang, B. Zhou, A note on sum of powers of the Laplacian eigenvalues of bipartite graphs. Lin. Algebra Appl. 430, 2503–2510 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  462. 462.
    A. Torgašev, Graphs whose energy does not exceed 3. Czech. Math. J. 36, 167–171 (1986)Google Scholar
  463. 463.
    V. Trevisan, J.B. Carvalho, R. Del-Vecchio, C. Vinagre, Laplacian energy of diameter 3 trees. Appl. Math. Lett. 24, 918–923 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  464. 464.
    L. Türker, An upper bound for total π-electron energy of alternant hydrocarbons. MATCH Commun. Math. Chem. 16, 83–94 (1984)Google Scholar
  465. 465.
    L. Türker, An approximate method for the estimation of total π-electron energies of alternant hydrocarbons. MATCH Commun. Math. Comput. Chem. 28, 261–276 (1992)Google Scholar
  466. 466.
    L. Türker, An approximate Hückel total π-electron energy formula for benzenoid aromatics. Polyc. Arom. Comp. 4, 107–114 (1994)CrossRefGoogle Scholar
  467. 467.
    L. Türker, A novel total π-electron energy formula for alternant hydrocarbons – Angle of total π-electron energy. MATCH Commun. Math. Comput. Chem. 30, 243–252 (1994)zbMATHGoogle Scholar
  468. 468.
    L. Türker, A novel approach to the estimation of total π-electron energies of cyclic alternant hydrocarbons. MATCH Commun. Math. Comput. Chem. 30, 253–268 (1994)zbMATHGoogle Scholar
  469. 469.
    L. Türker, A novel formula for the total π-electron energy of alternant hydrocarbons. MATCH Commun. Math. Comput. Chem. 32, 175–184 (1995)Google Scholar
  470. 470.
    L. Türker, Contemplation on the total π-electron energies of alternant hydrocarbons. MATCH Commun. Math. Comput. Chem. 32, 185–192 (1995)Google Scholar
  471. 471.
    L. Türker, Approximation of Hückel total π-electron energies of benzenoid hydrocarbons. ACH – Models Chem. 133, 407–414 (1996)Google Scholar
  472. 472.
    L. Türker, I. Gutman, Iterative estimation of total π-electron energy. J. Serb. Chem. Soc. 70, 1193–1197 (2005)CrossRefGoogle Scholar
  473. 473.
    P. van Mieghem, Graph Spectra for Complex Networks (Cambridge University Press, Cambridge, 2011), Section 7.8.2Google Scholar
  474. 474.
    S. Wagner, Energy bounds for graphs with fixed cyclomatic number. MATCH Commun. Math. Comput. Chem. 68, 661–674 (2012)Google Scholar
  475. 475.
    H.B. Walikar, I. Gutman, P.R. Hampiholi, H.S. Ramane, Non-hyperenergetic graphs. Graph Theor. Notes New York 41, 14–16 (2001)MathSciNetGoogle Scholar
  476. 476.
    H.B. Walikar, H.S. Ramane, Energy of some cluster graphs. Kragujevac J. Sci. 23, 51–62 (2001)Google Scholar
  477. 477.
    H.B. Walikar, H.S. Ramane, Energy of some bipartite cluster graphs. Kragujevac J. Sci. 23, 63–74 (2001)Google Scholar
  478. 478.
    H.B. Walikar, H.S. Ramane, P.R. Hampiholi, in On the Energy of a Graph, ed. by R. Balakrishnan, H.M. Mulder, A. Vijayakumar. Graph Connections (Allied, New Delhi, 1999), pp. 120–123Google Scholar
  479. 479.
    H.B. Walikar, H.S. Ramane, P.R. Hampiholi, in Energy of Trees with Edge Independence Number Three, ed. by R. Nadarajan, P.R. Kandasamy. Mathematical and Computational Models (Allied Publishers, New Delhi, 2001), pp. 306–312Google Scholar
  480. 480.
    D. Wang, H. Hua, Minimality considerations for graph energy over a class of graphs. Comput. Math. Appl. 56, 3181–3187 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  481. 481.
    H. Wang, H. Hua, Note on Laplacian energy of graphs. MATCH Commun. Math. Comput. Chem. 59, 373–380 (2008)MathSciNetzbMATHGoogle Scholar
  482. 482.
    M. Wang, H. Hua, D. Wang, Minimal energy on a class of graphs. J. Math. Chem. 44, 1389–1402 (2008)MathSciNetCrossRefGoogle Scholar
  483. 483.
    W. Wang, Ordering of Hückel trees according to minimal energies. Lin. Algebra Appl. 430, 703–717 (2009)zbMATHCrossRefGoogle Scholar
  484. 484.
    W.H. Wang, Ordering of unicyclic graphs with perfect matching by minimal energies. MATCH Commun. Math. Comput. Chem. 66, 927–942 (2011)MathSciNetGoogle Scholar
  485. 485.
    W. Wang, A. Chang, D. Lu, Unicyclic graphs possessing Kekulé structures with minimal energy. J. Math. Chem. 42, 311–320 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  486. 486.
    W. Wang, A. Chang, L. Zhang, D. Lu, Unicyclic Hückel molecular graphs with minimal energy. J. Math. Chem. 39, 231–241 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  487. 487.
    W. Wang, L. Kang, Ordering of the trees with a perfect matching by minimal energies. Lin. Algebra Appl. 431, 946–961 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  488. 488.
    W. Wang, L. Kang, Ordering of the trees by minimal energy. J. Math. Chem. 47, 937–958 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  489. 489.
    W.H. Wang, L. Kang, Ordering of unicyclic graphs by minimal energies and Hosoya indices. Util. Math., in pressGoogle Scholar
  490. 490.
    F. Wei, B. Zhou, N. Trinajstić, Minimal spectrum-sums of bipartite graphs with exactly two vertex-disjoint cycles. Croat. Chem. Acta 81, 363–367 (2008)Google Scholar
  491. 491.
    E.P. Wigner, Characteristic vectors of bordered matrices with infinite dimmensions. Ann. Math. 62, 548–564 (1955)MathSciNetzbMATHCrossRefGoogle Scholar
  492. 492.
    E.P. Wigner, On the distribution of the roots of certain symmetric matrices. Ann. Math. 67, 325–327 (1958)MathSciNetzbMATHCrossRefGoogle Scholar
  493. 493.
    J. Wishart, The generalized product moment distribution in samples from a normal multivariate population. Biometrika 20A, 32–52 (1928)Google Scholar
  494. 494.
    L. Xu, On biregular graphs whose energy exceeds the number of vertices. MATCH Commun. Math. Comput. Chem. 66, 959–970 (2011)MathSciNetGoogle Scholar
  495. 495.
    K. Xu, L. Feng, Extremal energies of trees with a given domination number. Lin. Algebra Appl. 435, 2382–2393 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  496. 496.
    L. Xu, Y. Hou, Equienergetic bipartite graphs. MATCH Commun. Math. Comput. Chem. 57, 363–370 (2007)MathSciNetzbMATHGoogle Scholar
  497. 497.
    W. Yan, L. Ye, On the minimal energy of trees with a given diameter. Appl. Math. Lett. 18, 1046–1052 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  498. 498.
    W. Yan, L. Ye, On the maximal energy and the Hosoya index of a type of trees with many pendent vertices. MATCH Commun. Math. Comput. Chem. 53, 449–459 (2005)MathSciNetzbMATHGoogle Scholar
  499. 499.
    W. Yan, Z. Zhang, Asymptotic energy of lattices. Physica A388, 1463–1471 (2009)MathSciNetGoogle Scholar
  500. 500.
    Y. Yang, B. Zhou, Minimal energy of bicyclic graphs of a given diameter. MATCH Commun. Math. Comput. Chem. 59, 321–342 (2008)MathSciNetzbMATHGoogle Scholar
  501. 501.
    Y. Yang, B. Zhou, Bipartite bicyclic graphs with large energies. MATCH Commun. Math. Comput. Chem. 61, 419–442 (2009)MathSciNetzbMATHGoogle Scholar
  502. 502.
    X. Yao, Maximum energy trees with one maximum and one second maximum degree vertex. MATCH Commun. Math. Comput. Chem. 64, 217–230 (2010)MathSciNetGoogle Scholar
  503. 503.
    K. Yates, Hückel Molecular Orbital Theory (Academic, New York, 1978)Google Scholar
  504. 504.
    L. Ye, The energy of a type of lattices. Appl. Math. Lett. 24, 145–148 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  505. 505.
    L. Ye, R. Chen, Ordering of trees with given bipartition by their energies and Hosoya indices. MATCH Commun. Math. Comput. Chem. 52, 193–208 (2004)zbMATHGoogle Scholar
  506. 506.
    L. Ye, X. Yuan, On the minimal energy of trees with a given number of pendant vertices. MATCH Commun. Math. Comput. Chem. 57, 193–201 (2007)MathSciNetzbMATHGoogle Scholar
  507. 507.
    Z. You, B. Liu, On hypoenergetic unicyclic and bicyclic graphs. MATCH Commun. Math. Comput. Chem. 61, 479–486 (2009)MathSciNetzbMATHGoogle Scholar
  508. 508.
    Z. You, B. Liu, I. Gutman, Note on hypoenergetic graphs. MATCH Commun. Math. Comput. Chem. 62, 491–498 (2009)MathSciNetzbMATHGoogle Scholar
  509. 509.
    A. Yu, M. Lu, F. Tian, On the spectral radius of graphs. Lin. Algebra Appl. 387, 41–49 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  510. 510.
    A. Yu, M. Lu, F. Tian, New upper bounds for the energy of graphs. MATCH Commun. Math. Comput. Chem. 53, 441–448 (2005)MathSciNetzbMATHGoogle Scholar
  511. 511.
    A. Yu, X. Lv, Minimal energy on trees with k pendent vertices. Lin. Algebra Appl. 418, 625–633 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  512. 512.
    A. Yu, F. Tian, On the spectral radius of unicyclic graphs. MATCH Commun. Math. Comput. Chem. 51, 97–109 (2004)MathSciNetzbMATHGoogle Scholar
  513. 513.
    G. Yu, The energy and spanning trees of the Aztec diamonds. Discr. Math. 311, 38–44 (2011)zbMATHCrossRefGoogle Scholar
  514. 514.
    B. Zhang, Remarks on minimal energies of unicyclic bipartite graphs. MATCH Commun. Math. Comput. Chem. 61, 487–494 (2009)MathSciNetGoogle Scholar
  515. 515.
    F. Zhang, Two theorems of comparison of bipartite graphs by their energy. Kexue Tongbao 28, 726–730 (1983)zbMATHGoogle Scholar
  516. 516.
    F. Zhang, Z. Lai, Three theorems of comparison of trees by their energy. Sci. Explor. 3, 12–19 (1983)MathSciNetGoogle Scholar
  517. 517.
    F. Zhang, H. Li, On acyclic conjugated molecules with minimal energies. Discr. Appl. Math. 92, 71–84 (1999)zbMATHCrossRefGoogle Scholar
  518. 518.
    F. Zhang, H. Li, On Maximal Energy Ordering of Acyclic Conjugated Molecules, ed. by P. Hansen, P. Fowler, M. Zheng. Discrete Mathematical Chemistry (American Mathematical Society, Providence, 2000), pp. 385–392Google Scholar
  519. 519.
    F. Zhang, Z. Li, L. Wang, Hexagonal chain with minimal total π-electron energy. Chem. Phys. Lett. 37, 125–130 (2001)CrossRefGoogle Scholar
  520. 520.
    F. Zhang, Z. Li, L. Wang, Hexagonal chain with maximal total π-electron energy. Chem. Phys. Lett. 37, 131–137 (2001)CrossRefGoogle Scholar
  521. 521.
    J. Zhang, On tricyclic graphs with minimal energies. preprint, 2006Google Scholar
  522. 522.
    J. Zhang, B. Zhou, Energy of bipartite graphs with exactly two cycles. Appl. Math. J. Chinese Univ., Ser. A 20, 233–238 (in Chinese) (2005)Google Scholar
  523. 523.
    J. Zhang, B. Zhou, On bicyclic graphs with minimal energies. J. Math. Chem. 37, 423–431 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  524. 524.
    J. Zhang, B. Zhou, On minimal energies of non-starlike trees with given number of pendent vertices. MATCH Commun. Math. Comput. Chem. 62, 481–490 (2009)MathSciNetzbMATHGoogle Scholar
  525. 525.
    Y. Zhang, F. Zhang I. Gutman, On the ordering of bipartite graphs with respect to their characteristic polynomials. Coll. Sci. Pap. Fac. Sci. Kragugevac 9, 9–20 (1988)MathSciNetzbMATHGoogle Scholar
  526. 526.
    P. Zhao, B. Zhao, X. Chen, Y. Bai, Two classes of chains with maximal and minimal total π-electron energy. MATCH Commun. Math. Comput. Chem. 62, 525–536 (2009)MathSciNetzbMATHGoogle Scholar
  527. 527.
    B. Zhou, On spectral radius of nonnegative matrics. Australas. J. Combin. 22, 301–306 (2000)MathSciNetzbMATHGoogle Scholar
  528. 528.
    B. Zhou, Energy of graphs. MATCH Commun. Math. Comput. Chem. 51, 111–118 (2004)zbMATHGoogle Scholar
  529. 529.
    B. Zhou, On the energy of a graph. Kragujevac J. Sci. 26, 5–12 (2004)Google Scholar
  530. 530.
    B. Zhou, Lower bounds for energy of quadrangle-free graphs. MATCH Commun. Math. Comput. Chem. 55, 91–94 (2006)MathSciNetzbMATHGoogle Scholar
  531. 531.
    B. Zhou, On the sum of powers of the Laplacian eigenvalues of graphs. Lin. Algebra Appl. 429, 2239–2246 (2008)zbMATHCrossRefGoogle Scholar
  532. 532.
    B. Zhou, New upper bounds for Laplacian energy. MATCH Commun. Math. Comput. Chem. 62, 553–560 (2009)MathSciNetzbMATHGoogle Scholar
  533. 533.
    B. Zhou, More on energy and Laplacian energy. MATCH Commun. Math. Comput. Chem. 64, 75–84 (2010)MathSciNetGoogle Scholar
  534. 534.
    B. Zhou, More upper bounds for the incidence energy. MATCH Commun. Math. Comput. Chem. 64, 123–128 (2010)MathSciNetGoogle Scholar
  535. 535.
    B. Zhou, I. Gutman, Further properties of Zagreb indices. MATCH Commun. Math. Comput. Chem. 54, 233–239 (2005)MathSciNetzbMATHGoogle Scholar
  536. 536.
    B. Zhou, I. Gutman, On Laplacian energy of graphs. MATCH Commun. Math. Comput. Chem. 57, 211–220 (2007)MathSciNetzbMATHGoogle Scholar
  537. 537.
    B. Zhou, I. Gutman, Nordhaus–Gaddum-type relations for the energy and Laplacian energy of graphs. Bull. Acad. Serbe Sci. Arts (Cl. Math. Natur.) 134, 1–11 (2007)Google Scholar
  538. 538.
    B. Zhou, I. Gutman, A connection between ordinary and Laplacian spectra of bipartite graphs. Lin. Multilin. Algebra 56, 305–310 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  539. 539.
    B. Zhou, I. Gutman, T. Aleksić, A note on Laplacian energy of graphs. MATCH Commun. Math. Comput. Chem. 60, 441–446 (2008)MathSciNetzbMATHGoogle Scholar
  540. 540.
    B. Zhou, I. Gutman, J.A. de la Peña, J. Rada, L. Mendoza, On the spectral moments and energy of graphs. MATCH Commun. Math. Comput. Chem. 57, 183–191 (2007)MathSciNetGoogle Scholar
  541. 541.
    B. Zhou, A. Ilić, On distance spectral radius and distance energy of graphs. MATCH Commun. Math. Comput. Chem. 64, 261–280 (2010)MathSciNetGoogle Scholar
  542. 542.
    B. Zhou, A. Ilić, On the sum of powers of Laplacian eigenvalues of bipartite graphs. Czech. Math. J. 60, 1161–1169 (2010)zbMATHCrossRefGoogle Scholar
  543. 543.
    B. Zhou, F. Li, On minimal energies of trees of a prescribed diameter. J. Math. Chem. 39, 465–473 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  544. 544.
    B. Zhou, H.S. Ramane, On upper bounds for energy of bipartite graphs. Indian J. Pure Appl. Chem. 39, 483–490 (2008)MathSciNetzbMATHGoogle Scholar
  545. 545.
    B. Zhou, N. Trinajstić, On the sum–connectivity matrix and sum-connectivity energy of (molecular) graphs. Acta Chim. Slov. 57, 513–517 (2010)Google Scholar
  546. 546.
    B.X. Zhu, The Laplacian-energy like of graphs. Appl. Math. Lett. 24, 1604–1607 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  547. 547.
    J. Zhu, Minimal energies of trees with given parameters. Lin. Algebra Appl. 436, 3120–3131 (2012)zbMATHCrossRefGoogle Scholar
  548. 548.
    B. D. Acharya, S. B. Rao, T. Singh, The minimum robust domination energy of a connected graph. AKCE Int. J. Graphs Combin. 4, 139–143 (2007)MathSciNetzbMATHGoogle Scholar
  549. 549.
    B. D. Acharya, S. B. Rao, P. Sumathi, V. Swaminathan, Energy of a set of vertices in a graph. AKCE Int. J. Graphs Combin. 4, 145–152 (2007)MathSciNetzbMATHGoogle Scholar
  550. 550.
    C. Adiga, A. Bayad, I. Gutman, A. S. Shrikanth, The minimum covering energy of a graph. Kragujevac J. Sci. 34, 39–56 (2012)Google Scholar
  551. 551.
    M. R. Ahmadi, R. Jahano–Nezhad, Energy and Wiener index of zero–divisor graphs. Iran. J. Math. Chem. 2, 45–51 (2011)Google Scholar
  552. 552.
    S. Alikhani, M. A. Iranmanesh. Energy of graphs, matroids and Fibonacci numbers. Iran. J. Math. Sci. Inf. 5(2), 55–60 (2010)MathSciNetGoogle Scholar
  553. 553.
    Ş. B. Bozkurt, C. Adiga, D. Bozkurt, On the energy and Estrada index of strongly quotient graphs. Indian J. Pure Appl. Math. 43, 25–36 (2012)MathSciNetCrossRefGoogle Scholar
  554. 554.
    Ş. B. Bozkurt, D. Bozkurt, Randić energy and Randić Estrada index of a graph. Europ. J. Pure Appl. Math. 5, 88–96 (2012)MathSciNetGoogle Scholar
  555. 555.
    A. Chang, B. Deng, On the Laplacian energy of trees with perfect matchings. MATCH Commun. Math. Comput. Chem. 68, 767–776 (2012)Google Scholar
  556. 556.
    K. C. Das, K. Xu, I. Gutman, Comparison between Kirchhoff index and the Laplacian–energy–like invariant. Lin. Algebra Appl. 436 3661–3671 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  557. 557.
    I. Gutman, Bounds for all graph energies. Chem. Phys. Lett. 528, 72–74 (2012)CrossRefGoogle Scholar
  558. 558.
    I. Gutman, Estimating the Laplacian–energy–like molecular structure descriptor. Z. Naturforsch. 67a, 403–406 (2012)Google Scholar
  559. 559.
    I. Gutman, B. Furtula, E. O. D. Andriantiana, M. Cvetić, More trees with large energy and small size. MATCH Commun. Math. Comput. Chem. 68, 697–702 (2012)Google Scholar
  560. 560.
    W. H. Haemers, Seidel switching and graph energy. MATCH Commun. Math. Comput. Chem. 68, 653–659 (2012)Google Scholar
  561. 561.
    H. B. Hua, On maximal energy and Hosoya index of trees without perfect matching. Bull. Austral. Math. Soc. 81, 47–57 (2010)zbMATHCrossRefGoogle Scholar
  562. 562.
    S. Ji, J. Li, An approach to the problem of the maximal energy of bicyclic graphs. MATCH Commun. Math. Comput. Chem. 68, 741–762 (2012)Google Scholar
  563. 563.
    T. A. Le, J. W. Sander, Extremal energies of integral circulant graphs via multiplicativity. Lin. Algebra Appl. 437, 1408–1421 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  564. 564.
    J. Liu, B. Liu, Generalization for Laplacian energy. Appl. Math. J. Chinese Univ. 24, 443–450 (2009)zbMATHCrossRefGoogle Scholar
  565. 565.
    Z. Liu, Energy, Laplacian energy and Zagreb index of line graph, middle graph and total graph. Int. J. Contemp. Math. Sci. 5, 895–900 (2010)MathSciNetzbMATHGoogle Scholar
  566. 566.
    B. Lv, K. Wang, The energy of Kneser graphs. MATCH Commun. Math. Comput. Chem. 68, 763–765 (2012)Google Scholar
  567. 567.
    J. Rada, Bounds for the energy of normal digraphs. Lin. Multilin. Algebra 60 323–332 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  568. 568.
    J. W. Sander, T. Sander, The maximal energy of classes of integral circulant graphs. Discr. Appl. Math. 160, 2015–2029 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  569. 569.
    H. Y. Shan, J. Y. Shao, L. Zhang, C. X. He, Proof of a conjecture on trees with large energy. MATCH Commun. Math. Comput. Chem. 68, 703–720 (2012)Google Scholar
  570. 570.
    H. Y. Shan, J. Y. Shao, L. Zhang, C. X. He, A new method of comparing the energies of subdivision bipartite graphs. MATCH Commun. Math. Comput. Chem. 68, 721–740 (2012)Google Scholar
  571. 571.
    Y. Z. Song, P. Arbelaez, P. Hall, C. Li, A. Balikai, in Finding Semantic Structures in Image Hierarchies Using Laplacian Graph Fnergy, ed by K. Daniilidis, P. Maragos, N. Paragios, Computer Vision – CECV 2010 (European Conference on Computer Vision, 2010), Part IV, (Springer, Berlin, 2010), pp. 694–707Google Scholar
  572. 572.
    T. Tamizh Chelvam, S. Raja, I. Gutman, Strongly regular integral circulant graphs and their energies. Bull. Int. Math. Virt. Inst. 2, 9–16 (2012)Google Scholar
  573. 573.
    J. Zhang, J. Li, New results on the incidence energy of graphs. MATCH Commun. Math. Comput. Chem. 68, 777–803 (2012)Google Scholar
  574. 574.
    J. Zhu, Minimal energies of trees with given parameters. Lin. Algebra Appl. 436, 3120–3131 (2012).zbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  • Xueliang Li
    • 1
  • Yongtang Shi
    • 1
  • Ivan Gutman
    • 2
  1. 1.Center for CombinatoricsNankai UniversityTianjinChina
  2. 2.Faculty of ScienceUniversity of KragujevacKragujevacSerbia

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