Journal of Mathematical Chemistry

, Volume 46, Issue 1, pp 214–230 | Cite as

Chemical trees minimizing energy and Hosoya index

  • Clemens Heuberger
  • Stephan G. Wagner
Original Paper


The energy of a molecular graph is a popular parameter that is defined as the sum of the absolute values of a graph’s eigenvalues. It is well known that the energy is related to the matching polynomial and thus also to the Hosoya index via a certain Coulson integral. It is quite a natural problem to minimize the energy of trees with bounded maximum degree—clearly, the case of maximum degree 4 (so-called chemical trees) is the most important one. We will show that the trees with given maximum degree that minimize the energy are the same that have been shown previously to minimize the Hosoya index and maximize the Merrifield-Simmons index, thus also proving a conjecture due to Fischermann et al. Finally, we show that the minimum energy grows linearly with the size of the trees, with explicitly computable growth constants that only depend on the maximum degree.


Energy of graphs Matchings Chemical trees Matching polynomial Hosoya index 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Cvetković D.M., Doob M., Sachs H.: Spectra of Graphs, 3rd edn. Johann Ambrosius Barth, Heidelberg (1995)Google Scholar
  2. 2.
    Fischermann M., Gutman I., Hoffmann A., Rautenbach D., Vidović D., Volkmann L.: Extremal chemical trees. Z. Naturforsch. 57a, 49–52 (2002)Google Scholar
  3. 3.
    Gutman I.: Acyclic systems with extremal Huckel pi-electron energy. Theor. Chim. Acta 45, 79–87 (1977)CrossRefGoogle Scholar
  4. 4.
    I. Gutman, The energy of a graph. Ber. Math.-Statist. Sekt. Forsch. Graz. 100–105, Ber. No. 103, 22 (1978)Google Scholar
  5. 5.
    I. Gutman, The energy of a graph: old and new results, in Algebraic Combinatorics and Applications (Göβweinstein, 1999) (Springer, Berlin, 2001), pp. 196–211Google Scholar
  6. 6.
    Gutman I., Polansky O.E.: Mathematical Concepts in Organic Chemistry. Springer-Verlag, Berlin (1986)Google Scholar
  7. 7.
    Gutman I., Radenković S., Li N., Li S.: Extremal energy trees. MATCH Commun. Math. Comput. Chem. 59(2), 315–320 (2008)Google Scholar
  8. 8.
    C. Heuberger, H. Prodinger, S. Wagner, Positional number systems with digits forming an arithmetic progression. Monatsh. Math. (2008) doi: 10.1007/s00605-008-0008-8
  9. 9.
    Heuberger C., Wagner S.: Maximizing the number of independent subsets over trees with bounded degree. J. Graph Theory 58(1), 49–68 (2008)CrossRefGoogle Scholar
  10. 10.
    C. Heuberger, S.G. Wagner, On a class of extremal trees for various indices. Report 2008–15, Graz University of Technology, (2008)
  11. 11.
    Hoggatt V.E. Jr., Bicknell M.: Roots of Fibonacci polynomials. Fibonacci Quart. 11(3), 271–274 (1973)Google Scholar
  12. 12.
    H. Hosoya, Topological Index as a Common Tool for Quantum Chemistry, Statistical Mechanics, Graph Theory, in Mathematical and Computational Concepts in Chemistry (Dubrovnik, 1985), Ellis Horwood Ser. Math. Appl. (Horwood, Chichester, 1986), pp. 110–123Google Scholar
  13. 13.
    Hou Y.: Unicyclic graphs with minimal energy. J. Math. Chem. 29(3), 163–168 (2001)CrossRefGoogle Scholar
  14. 14.
    Hua H.: On minimal energy of unicyclic graphs with prescribed girth and pendent vertices. MATCH Commun. Math. Comput. Chem. 57(2), 351–361 (2007)Google Scholar
  15. 15.
    Hua H., Wang M.: Unicyclic graphs with given number of pendent vertices and minimal energy. Linear Algebra Appl. 426(2–3), 478–489 (2007)CrossRefGoogle Scholar
  16. 16.
    Li F., Zhou B.: Minimal energy of bipartite unicyclic graphs of a given bipartition. MATCH Commun. Math. Comput. Chem., 54(2), 379–388 (2005)Google Scholar
  17. 17.
    Li H.: On minimal energy ordering of acyclic conjugated molecules. J. Math. Chem. 25(2–3), 145–169 (1999)CrossRefGoogle Scholar
  18. 18.
    Li N., Li S.: On the extremal energies of trees. MATCH Commun. Math. Comput. Chem. 59(2), 291–314 (2008)Google Scholar
  19. 19.
    Li S., Li X., Zhu Z.: On tricyclic graphs with minimal energy. MATCH Commun. Math. Comput. Chem. 59(2), 397–419 (2008)Google Scholar
  20. 20.
    Li S., Li X., Zhu Z.: On minimal energy and Hosoya index of unicyclic graphs. MATCH Commun. Math. Comput. Chem. 61(2), 325–339 (2009)Google Scholar
  21. 21.
    Merrifield R.E., Simmons H.E.: Topological Methods in Chemistry. Wiley, New York (1989)Google Scholar
  22. 22.
    Wang W.-H., Chang A., Zhang L.-Z., Lu D.-Q.: Unicyclic Hückel molecular graphs with minimal energy. J. Math. Chem. 39(1), 231–241 (2006)CrossRefGoogle Scholar
  23. 23.
    Yan W., Ye L.: On the minimal energy of trees with a given diameter. Appl. Math. Lett. 18(9), 1046–1052 (2005)CrossRefGoogle Scholar
  24. 24.
    Yang Y., Zhou B.: Minimum energy of bicyclic graphs of a given diameter. MATCH Commun. Math. Comput. Chem. 59(2), 321–342 (2008)Google Scholar
  25. 25.
    Ye L., Yuan X.: On the minimal energy of trees with a given number of pendent vertices. MATCH Commun. Math. Comput. Chem. 57(1), 193–201 (2007)Google Scholar
  26. 26.
    Yu A., Lv X.: Minimum energy on trees with k pendent vertices. Linear Algebra Appl. 418(2–3), 625–633 (2006)CrossRefGoogle Scholar
  27. 27.
    Zhang F., Li H.: On acyclic conjugated molecules with minimal energies. Discrete Appl. Math. 92(1), 71–84 (1999)CrossRefGoogle Scholar
  28. 28.
    Zhou B., Li F.: On minimal energies of trees of a prescribed diameter. J. Math. Chem. 39(3–4), 465–473 (2006)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  1. 1.Institut für Mathematik BTechnische Universität GrazGrazAustria
  2. 2.Department of Mathematical SciencesUniversity of StellenboschStellenboschSouth Africa

Personalised recommendations