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CMMSE-on the first general Zagreb index

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Abstract

The aim of this paper is to obtain new inequalities involving the first general Zagreb index, and characterize graphs which are extremal with respect to them. We also obtain inequalities involving the forgotten and second general Zagreb indices.

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References

  1. H. Abdo, D. Dimitrov, I. Gutman, On extremal trees with respect to the F-index. Kuwait J. Sci. 44(3), 1–8 (2017)

    Google Scholar 

  2. C. Alsina, When Less is More: Visualizing Basic Inequalities (Mathematical Association of America, Washington, DC, 2009)

    Google Scholar 

  3. V. Andova, M. Petrusevski, Variable Zagreb Indices and Karamata’s inequality. MATCH Commun. Math. Comput. Chem. 65, 685–690 (2011)

    Google Scholar 

  4. G. Britto Antony Xavier, E. Suresh, I. Gutman, Counting relations for general Zagreb indices. Kragujevac J. Math. 38, 95–103 (2014)

    Article  Google Scholar 

  5. Z. Che, Z. Chen, Lower and upper bounds of the forgotten topological index. MATCH Commun. Math. Comput. Chem. 76, 635–648 (2016)

    Google Scholar 

  6. N. De, S.M.A. Nayeem, A. Pal, F-index of some graph operations. Discrete Math. Algorithms Appl. (2016). doi:10.1142/S1793830916500257

    Article  Google Scholar 

  7. N. De, S.M.A. Nayeem, A. Pal, The F-coindex of some graph operations. SpringerPlus 5, 221 (2016). doi:10.1186/s40064-016-1864-7

    Article  PubMed  PubMed Central  Google Scholar 

  8. S. Fajtlowicz, On conjectures of Graffiti-II. Congr. Numer. 60, 187–197 (1987)

    Google Scholar 

  9. B. Furtula, I. Gutman, A forgotten topological index. J. Math. Chem. 53(4), 1184–1190 (2015)

    Article  CAS  Google Scholar 

  10. M. Ghorbani, M. Songhori, I. Gutman, Modified Narumi-Katayama index. Kragujevac J. Sci. 34, 57–64 (2012)

    Google Scholar 

  11. I. Gutman, B. Furtula (eds.), Recent Results in the Theory of Randić Index (University of Kragujevac, Kragujevac, 2008)

    Google Scholar 

  12. I. Gutman, J. Tosovic, Testing the quality of molecular structure descriptors. Vertex-degreebased topological indices. J. Serb. Chem. Soc. 78(6), 805–810 (2013)

    Article  CAS  Google Scholar 

  13. I. Gutman, N. Trinajstić, Graph theory and molecular orbitals. Total \(\pi \)-electron energy of alternant hydrocarbons. Chem. Phys. Lett. 17, 535–538 (1972)

    Article  CAS  Google Scholar 

  14. G.H. Hardy, J.E. Littlewood, G. Polya, Inequalities (Cambridge Univ. Press, Cambridge, 1952)

    Google Scholar 

  15. X. Li, I. Gutman, Mathematical Aspects of Randić Type Molecular Structure Descriptors (University Kragujevac, Kragujevac, 2006)

    Google Scholar 

  16. X. Li, H. Zhao, Trees with the first smallest and largest generalized topological indices. MATCH Commun. Math. Comput. Chem. 50, 57–62 (2004)

    Google Scholar 

  17. X. Li, J. Zheng, A unified approach to the extremal trees for different indices. MATCH Commun. Math. Comput. Chem. 54, 195–208 (2005)

    CAS  Google Scholar 

  18. M. Liu, B. Liu, Some properties of the first general Zagreb index. Australas. J. Comb. 47, 285–294 (2010)

    Google Scholar 

  19. A. Miličević, S. Nikolić, On variable Zagreb indices. Croat. Chem. Acta 77, 97–101 (2004)

    Google Scholar 

  20. H. Narumi, M. Katayama, Simple topological index. A newly devised index characterizing the topological nature of structural isomers of saturated hydrocarbons. Mem. Fac. Eng. Hokkaido Univ. 16, 209–214 (1984)

    CAS  Google Scholar 

  21. S. Nikolić, G. Kovačević, A. Miličević, N. Trinajstić, The Zagreb Indices 30 years after. Croat. Chem. Acta 76, 113–124 (2003)

    Google Scholar 

  22. S. Nikolić, A. Miličević, N. Trinajstić, A. Jurić, On use of the variable Zagreb \(^\nu M_2\) index in QSPR: boiling points of benzenoid hydrocarbons. Molecules 9, 1208–1221 (2004)

    Article  PubMed  Google Scholar 

  23. M. Randić, On characterization of molecular branching. J. Am. Chem. Soc. 97, 6609–6615 (1975)

    Article  Google Scholar 

  24. M. Randić, Novel graph theoretical approach to heteroatoms in QSAR. Chemom. Intel. Lab. Syst. 10, 213–227 (1991)

    Article  Google Scholar 

  25. M. Randić, On computation of optimal parameters for multivariate analysis of structure-property relationship. J. Chem. Inf. Comput. Sci. 31, 970–980 (1991)

    Article  Google Scholar 

  26. M. Randić, D. Plavšić, N. Lerš, Variable connectivity index for cycle-containing structures. J. Chem. Inf. Comput. Sci. 41, 657–662 (2001)

    Article  CAS  PubMed  Google Scholar 

  27. J.M. Rodríguez, J.M. Sigarreta, New results on the harmonic index and its generalizations. MATCH Commun. Math. Comput. Chem. 78(2), 387–404 (2017)

    Google Scholar 

  28. J.A. Rodríguez-Velázquez, J.M. Sigarreta, On the Randić index and condicional parameters of a graph. MATCH Commun. Math. Comput. Chem. 54, 403–416 (2005)

    Google Scholar 

  29. J.A. Rodríguez-Velázquez, J. Tomás-Andreu, On the Randić index of polymeric networks modelled by generalized Sierpinski graphs. MATCH Commun. Math. Comput. Chem. 74, 145–160 (2015)

    Google Scholar 

  30. J.M. Sigarreta, Bounds for the geometric-arithmetic index of a graph. Miskolc Math. Notes 16, 1199–1212 (2015)

    Article  Google Scholar 

  31. M. Singh, K.C. Das, S. Gupta, A.K. Madan, Refined variable Zagreb indices: highly discriminating topological descriptors for QSAR/QSPR. Int. J. Chem. Model. 6(2–3), 403–428 (2014)

    Google Scholar 

  32. L.A. Székely, L.H. Clark, R.C. Entringer, An inequality for degree sequences. Discrete Math. 103, 293–300 (1992)

    Article  Google Scholar 

  33. M. Vöge, A.J. Guttmann, I. Jensen, On the number of benzenoid hydrocarbons. J. Chem. Inf. Comput. Sci. 42, 456–466 (2002)

    Article  CAS  PubMed  Google Scholar 

  34. H. Wiener, Structural determination of paraffin boiling points. J. Am. Chem. Soc. 69, 17–20 (1947)

    Article  CAS  PubMed  Google Scholar 

  35. S. Zhang, W. Wang, T.C.E. Cheng, Bicyclic graphs with the first three smallest and largest values of the first general Zagreb index. MATCH Commun. Math. Comput. Chem. 55, 579–592 (2006)

    Google Scholar 

  36. H. Zhang, S. Zhang, Unicyclic graphs with the first three smallest and largest values of the first general Zagreb index. MATCH Commun. Math. Comput. Chem. 55, 427–438 (2006)

    CAS  Google Scholar 

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Acknowledgements

This work is supported in part by two grants from Ministerio de Economía y Competititvidad, Agencia Estatal de Investigación (AEI) and Fondo Europeo de Desarrollo Regional (FEDER) (MTM 2016-78227-C2-1-P and MTM 2015-69323-REDT), Spain, and a grant from CONACYT (FOMIX-CONACyT-UAGro 249818), México.

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Authors and Affiliations

  1. Departamento de Matemáticas, Universidad Carlos III de Madrid, Avenida de la Universidad 30, 28911, Leganés, Madrid, Spain

    José M. Rodríguez

  2. Facultad de Matemáticas, Universidad Autónoma de Guerrero, Carlos E. Adame No.54 Col. Garita, 39650, Acapulco, Gro, Mexico

    José L. Sánchez & José M. Sigarreta

  3. Universidad de Holguín, Holguín, Cuba

    José L. Sánchez

Authors
  1. José M. Rodríguez
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  2. José L. Sánchez
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  3. José M. Sigarreta
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Correspondence to José M. Rodríguez.

Additional information

This is one of several papers published together in Journal of Mathematical Chemistry on the “Special Issue: CMMSE 2017”.

J. M. Rodríguez, J. L. Sánchez and J. M. Sigarreta: Supported in part by a grant from CONACYT (FOMIX-CONACyT-UAGro 249818), México.

J. M. Rodríguez and J. M. Sigarreta: Supported in part by two grants from Ministerio de Economía y Competititvidad, Agencia Estatal de Investigación (AEI) and Fondo Europeo de Desarrollo Regional (FEDER) (MTM 2016-78227-C2-1-P and MTM 2015-69323-REDT), Spain.

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Rodríguez, J.M., Sánchez, J.L. & Sigarreta, J.M. CMMSE-on the first general Zagreb index. J Math Chem 56, 1849–1864 (2018). https://doi.org/10.1007/s10910-017-0816-y

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  • DOI: https://doi.org/10.1007/s10910-017-0816-y

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