Advertisement

Graphs having extremal monotonic topological indices with bounded vertex k-partiteness

Original Research
  • 40 Downloads

Abstract

The vertex k-partiteness \(v_k(G)\) of graph G is defined as the fewest number of vertices whose deletion from G yields a k-partite graph. In this paper, we introduce some monotonic topological indices, and characterize the extremal corresponding graphs among graphs of order n and fixed vertex k-partiteness.

Keywords

Vertex k-partiteness Monotonic topological index Extremal graph 

Mathematics Subject Classification

05C12 05C35 

Notes

Acknowledgements

The authors would like to express their sincere gratitude to the editors and the anonymous referees for many friendly and helpful suggestions, which led to great deal of improvement of the original manuscript.

References

  1. 1.
    Alizadeh, Y., Iranmanesh, A., Dos̆lić, T.: Additively weighted Harary index of some composite graphs. Discrete Math. 313, 26–34 (2013)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Ashrafi, A.R., Saheli, M., Ghorbani, M.: The eccentric connectivity index of nanotubes and nanotori. J. Comput. Appl. Math. 235, 4561–4566 (2011)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Bondy, J.A., Murty, U.S.R.: Graph Theory with Applications. Macmillan and Elsevier, London and New York (1976)CrossRefMATHGoogle Scholar
  4. 4.
    de Freitas, M.A.A., Gutman, I., Robbiano, M.: Graphs with maximum Laplacian-energy-like invariant and incidence energy. MATCH Commun. Math. Comput. Chem. 75, 331–342 (2016)MathSciNetMATHGoogle Scholar
  5. 5.
    Fallat, S., Fan, Y.Z.: Bipartiteness and the least eigenvalue of signless Laplacian of graphs. Linear Algebra Appl. 436, 3254–3267 (2012)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Furtula, B., Gutman, I., Dehmer, M.: On structure-sensitivity of degree-based topological indices. Appl. Math. Comput. 219, 8973–8978 (2013)MathSciNetMATHGoogle Scholar
  7. 7.
    Gupta, S., Singh, M., Madan, A.K.: Connective eccentricity index: a novel topological descriptor for predicting biological activity. J. Mol. Graph. Model. 18, 18–25 (2000)CrossRefGoogle Scholar
  8. 8.
    Gupta, S., Singh, M., Madan, A.K.: Application of graph theory: relationship of eccentric connectivity index and Wiener’s index with anti-inflammatory activity. J. Math. Anal. Appl. 266, 259–268 (2002)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Gutman, I., Rus̆c̆ić, B., Trinajstić, N., Wilcox, C.F.: Graph theory and molecular orbitals. J. Chem. Phys. 62, 3390–3405 (1975)CrossRefGoogle Scholar
  10. 10.
    Gutman, I., Trinajstić, N.: Graph theory and molecular orbitals. Total $\pi $-electron energy of alternant hydrocarbons. Chem. Phys. Lett. 17, 535–538 (1972)CrossRefGoogle Scholar
  11. 11.
    Gutman, I.: Degree-based topological indices. Croat. Chem. 86, 351–361 (2013)CrossRefGoogle Scholar
  12. 12.
    Gutman, I., Medina, L.C., Pizarro, P., Robbiano, M.: Graphs with maximum Laplacian and signless Laplacian Estrada index. Discrete Appl. Math. 339, 2664–2671 (2016)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Gutman, I., Tos̆ović, J.: Testing the quality of molecular structure descriptors VertexCdegree-based topological indices. J. Serb. Chem. Soc. 78, 805–810 (2013)CrossRefGoogle Scholar
  14. 14.
    Hua, H.B., Zhang, S.G.: On the reciprocal degree distance of graphs. Discrete Appl. Math. 160, 1152–1163 (2012)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Li, H.S., Li, S.C., Zhang, H.H.: On the maximal connective eccentricity index of bipartite graphs with some given parameters. J. Math. Anal. Appl. 454, 453–467 (2017)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Liu, J.B., Pan, X.F.: Minimizing Kirchhoff index among graphs with a given vertex bipartiteness. Appl. Math. Comput. 291, 84–88 (2016)MathSciNetGoogle Scholar
  17. 17.
    Nihat, A., Das, K.C., Çevik, A.S.: Some properties on the tensor product of graphs obtained by monogenic semigroups. Appl. Math. Comput. 235, 352–357 (2014)MathSciNetMATHGoogle Scholar
  18. 18.
    Plavs̆ić, D., Nikolić, S., Trinajstić, N., Mihalić, Z.: On the Harary index for the characterization of chemical graphs. J. Math. Chem. 12, 235–250 (1993)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Robbiano, M., Morales, K.T., San Martín, B.: Extremal graphs with bounded vertex bipartiteness number. Linear Algebra Appl. 493, 28–36 (2016)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Sardana, S., Madan, A.K.: Predicting anti-HIV activity of TIBO derivatives: a computational approach using a novel topological descriptor. Mol. Model 8, 258–265 (2002)CrossRefGoogle Scholar
  21. 21.
    Todeschini, R., Conaoni, V.: New local vertex invariants and molecular descriptors based on funtions of the vertex degrees. Math. Comput. 64, 359–372 (2010)Google Scholar
  22. 22.
    Wiener, H.: Structural determination of paraffin boiling point. J. Am. Chem. Soc. 69, 17–20 (1947)CrossRefGoogle Scholar
  23. 23.
    Xu, K., Das, K.C., Trinajstić, N.: The Harary Index of a Graph. Springer, Heidelberg (2015)CrossRefMATHGoogle Scholar
  24. 24.
    Xu, K., Liu, M.H., Das, K.C., Gutman, I., Furtula, B.: A survey on graphs extremal with respect to distance-based topological indices. Math. Comput. Chem. 71, 461–508 (2014)MathSciNetMATHGoogle Scholar
  25. 25.
    Xu, K., Das, K.C., Liu, H.: Some extremal results on the connective eccentricity index of graphs. J. Math. Anal. Appl. 433, 803–817 (2016)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Xu, K., Alizadeh, Y., Das, K.C.: On two eccentricity-based topological indices of graphs. Discrete Appl. Math. (2017).  https://doi.org/10.1016/j.dam.2017.08.010MathSciNetMATHGoogle Scholar

Copyright information

© Korean Society for Computational and Applied Mathematics 2018

Authors and Affiliations

  1. 1.School of Mathematics and Computer ScienceChizhou UniversityChizhouChina
  2. 2.School of Mathematics and PhysicsAnhui Jianzhu UniversityHefeiChina

Personalised recommendations