On leap Zagreb indices of graphs

Document Type : Original paper

Authors

1 University of Kragujevac

2 Department of Mathematics, University of Mysore, Mysusu, India

3 Department of Mathematics, University of Mysore, Mysuru, India

Abstract

The first and second Zagreb indices of a graph are equal, respectively, to the sum of squares of the vertex degrees,
and the sum of the products of the degrees of pairs of adjacent vertices. We now consider analogous graph invariants, based on the second degrees of vertices (number of their second neighbors), called leap Zagreb indices. A number of their basic properties is established.

Keywords

Main Subjects

References

[1] A. Ashrafi, T. Došlić, and A. Hamzeh, The Zagreb coindices of graph operations, Discrete Appl. Math. 158 (2010), no. 15, 1571–1578.
[2] B. Basavanagoud and S. Patil, A note on hyper-Zagreb index of graph operations, Iran. J. Math. Chem. 7 (2016), no. 1, 89–92.
[3] B. Borovicanin, K.C. Das, B. Furtula, and I. Gutman, Bounds for Zagreb indices, MATCH Commun. Math. Comput. Chem 78 (2017), no. 1, 17–100.
[4] R.M. Damerell, On Moore graphs, Proc. Cambridge Phil. Soc. 74 (1973), no. 2, 227–236.
[5] K.C. Das and I. Gutman, Some properties of the second Zagreb index, MATCH Commun. Math. Comput. Chem 52 (2004), no. 1, 103–112.
[6] T. Došlić, Vertex-weighted Wiener polynomials for composite graphs, Ars Math. Contemp. 1 (2008), no. 1, 66–80.
[7] M. Eliasi, A. Iranmanesh, and I. Gutman, Multiplicative versions of first Zagreb index, MATCH Commun. Math. Comput. Chem. 68 (2012), no. 1, 217–230.
[8] B. Furtula, I. Gutman, and M. Dehmer, On structure-sensitivity of degree-based topological indices, Appl. Math. Comput. 219 (2013), no. 17, 8973–8978.
[9] I. Gutman, Multiplicative Zagreb indices of trees, Bull. Soc. Math. Banja Luka 18 (2011), 17–23.
[10] I. Gutman, Degree-based topological indices, Croat. Chem. Acta 86 (2013), no. 4, 351–361.
[11] I. Gutman and K.C. Das, The first Zagreb index 30 years after, MATCH Commun. Math. Comput. Chem 50 (2004), 83–92.
[12] I. Gutman, B. Furtula, K. Vukićević, and G. Popivoda, On Zagreb indices and coindices, MATCH Commun. Math. Comput. Chem 74 (2015), no. 1, 5–16.
[13] I. Gutman, B. Ruščić, N. Trinajstić, and C.F. Wilcox, Graph theory and molecular orbitals. XII. acyclic polyenes, J. Chem. Phys. 62 (1975), no. 9, 3399–3405.
[14] I. Gutman and N. Trinajstić, Graph theory and molecular orbitals. Total ϕelectron energy of alternant hydrocarbons, Chem. Phys. Lett. 17 (1972), no. 4, 535–538.
[15] F. Harary, Graph Theory, Addison-Wesley, Reading, 1969.
[16] A. Ilić and D. Stevanović, On comparing Zagreb indices, MATCH Commun. Math. Comput. Chem. 62 (2009), no. 3, 681–687.
[17] A. Ilić and B. Zhou, On reformulated Zagreb indices, Discrete Appl. Math. 160 (2012), no. 3, 204–209.
[18] M.H. Khalifeh, H. Yousefi-Azari, and A. Ashrafi, The first and second Zagreb indices of some graph operations, Discrete Appl. Math. 157 (2009), no. 4, 804–811.
[19] A. Miličević, S. Nikolić, and N. Trinajstić, On reformulated Zagreb indices, Mol.
Diversity 8 (2004), no. 4, 393–399.
[20] S. Nikolić, G. Kovačević, A. Miličević, and N. Trinajstić, The Zagreb indices 30 years after, Croat. Chem. Acta 76 (2003), no. 2, 113–124.
[21] K. Pattabiraman and M. Vijayaragavan, Hyper Zagreb indices and its coindices of graphs, Bull. Int. Math. Virt. Inst. 7 (2017), no. 1, 31–41.
[22] N.D. Soner and A.M. Naji, The k-distance neighborhood polynomial of a graph, Int. J. Math. Comput. Sci. WASET Conference Proceedings, San Francico, USA, Sep 26-27, 3 (2016), no. 9, part XV, 2359–2364.
[23] K. Xu and K.C. Das, Trees, unicyclic, and bicyclic graphs extremal with respect to multiplicative sum Zagreb index, MATCH Commun. Math. Comput. Chem. 68 (2012), no. 1, 257–272.
[24] K. Xu, K.C. Das, and K. Tang, On the multiplicative Zagreb coindex of graphs, Opuscula Math. 33 (2013), no. 1, 191–204.
[25] K. Xu and H. Hua, A unified approach to extremal multiplicative Zagreb indices for trees, unicyclic and bicyclic graphs, MATCH Commun. Math. Comput. Chem. 68 (2012), no. 1, 241–256.
[26] S. Yamaguchi, Estimating the Zagreb indices and the spectral radius of triangleand quadrangle-free connected graphs, Chem. Phys. Lett. 458 (2008), no. 4, 396–398.
Communications in Combinatorics and Optimization
Volume 2, Issue 2
September 2017
Pages 99-117

Files

Share

How to cite

Statistics