On the Graovac-Ghorbani Index

Document Type : Research Paper

Authors

1 Department of mathematics, Shahid Rajaee Teacher Training University

2 Department of Mathematics, SRTT University

3 Actinum Chemical Research, Italy

Abstract

For the edge e = uv of a graph G, let nu = n(u|G) be the number of vertices of G lying closer to the vertex u than to the vertex v and nv= n(v|G) can be defined simailarly. Then the ABCGG index of G is defined as ABCGG =\sum_{e=uv} \sqrt{f(u,v)}, where f(u,v)= (nu+nv-2)/nunvThe aim of this paper is to give some new results on this graph invariant. We also calculate the ABCGG of an infinite family of fullerenes.

Keywords

Main Subjects


  1.  

    1. D. Dimitrov, B. Ikica and R. Škrekovski, Remarks on the Graovac-Ghorbani index of bipartite graphs, arXiv:1609.01406v1.
    2. E. Estrada, L. Torres, L. Rodríguez and I. Gutman, An atom-bond connectivity index: Modelling the enthalpy of formation of alkanes, Indian J. Chem. 37A (1998) 849−855.
    3. E. Estrada, Atom-bond connectivity and the energetic of branched alkanes, Chem. Phys. Lett. 463 (2008) 422−425.
    4. B. Furtula, I. Gutman and K. C. Das, On atom-bond molecule structure descriptors, J. Serb. Chem. Soc. 80 (2015) 1−7.
    5. B. Furtula and I. Gutman, in: I. Gutman, B. Furtula (Eds.) Novel Molecular Structure Descriptors − Theory and Applications, Vol. I, University of Kragujevac, Kragujevac, 2010, pp. 137−172.
    6. B. Furtula, Atom-bond connectivity index versus Graovac-Ghorbani analog, MATCH Commun. Math. Comput. Chem. 75 (2016) 233−242.
    7. M. Ghorbani, Enumeration of heterofullerenes: A survey, MATCH Commun. Math. Comput. Chem. 68 (2012) 381−414.
    8. M. Ghorbani and Sh. Rahmani, Study of Mostar index of fullerene graphs, Iranian J. Math. Sci. Inf., in press.
    9. A. Graovac and M. Ghorbani, A new version of the atom-bond connectivity index, Acta Chim. Slov. 57 (2010) 609−612.
    10. I. Gutman and A. A. Dobrynin, The Szeged index − a success story, Graph Theory Notes New York 34 (1998) 37−44.
    11. M. H. Khalifeh, H. Yousefi-Azari and A. R. Ashrafi, Vertex and edge PI indices of Cartesian product graphs, Discrete Appl. Math. 156 (2008) 1780−1789.
    12. H. W. Kroto, R. J. Heath, S. C. O’Brien, R. F. Curl and R. E. Smalley, C60: Buckminsterfullerene, Nature 318 (1985) 162−163.
    13. H. W. Kroto, J. E. Fichier and D. E. Cox, The Fulerene, Pergamon Press, New York, 1993.
    14. R. Todeschini and V. Consonni, Handbook of Molecular Descriptors, Wiley-VCH, Weinheim, 2000.
    15. N. Trinajstić, Chemical Graph Theory, 2nd revised edn, CRC Press, Boca Raton, FL, 1992.
    16. N. Trinajstić and I. Gutman, Mathematical Chemistry, Croat. Chem. Acta. 75 (2002) 329 – 356.
    17. H. Wiener, Structural determination of the paraffin boiling points, J. Am. Chem. Soc. 69 (1947) 17−20.