On Vertex-Edge Eccentric Connectivity Index of Wheel related and
Windmill Graphs

Shiladhar Pawar

doi:https://doi.org/10.14445/22315373/IJMTT-V68I4P508

International Journal of Mathematics Trends and Technology

Research Article | Open Access | Download PDF

Volume 68 | Issue 4 | Year 2022 | Article Id. IJMTT-V68I4P508 | DOI : https://doi.org/10.14445/22315373/IJMTT-V68I4P508

On Vertex-Edge Eccentric Connectivity Index of Wheel related and Windmill Graphs

Shiladhar Pawar

Received	Revised	Accepted
13 Mar 2022	15 Apr 2022	19 Apr 2022

Abstract

In a graph G = (V, E), a vertex v∈V, ve-dominates every edge incident to it as well as every edge adjacent to these incident edges. The vertex-edge degree of a vertex v, is denoted by d<sup>ve</sup>(v) and is the number of edges ve-dominated by v. In this paper, we introduce the vertex-edge eccentric connectivity index of a graph G, denoted by ξ^c_vee(G) and is equal to the sum of the product of the connectivity and ve-degree of the vertices of G. We calculate the vertex-edge eccentric connectivity index of some wheel related graphs and windmill graphs. Finally, we obtain some upper and lower bounds on ξ^c_vee(G).

Keywords

ve-Degree of Vertex, Connectivity, Eccentricity Index.

References

[1] B. Borovicanin, K. C. Das, B. Furtula, and I. Gutman, Bounds for Zagreb indices, MATCH Commun. Math. Comput. Chem., 78(1) (2017), 17-100.
[2] R. Boutrig, M. Chellali, T. W. Haynes, and S. T. Hedetniemi, Vertex-edge domination in graphs, Aequationes Math. 90(2) (2016), 355-366.
[3] M. Chellali, T.W. Haynes, S.T. Hedetniemi, T.M. Lewis, On ve-degrees and ev-degrees in graphs, Discrete Mathematics 340 (2017), 31-38.
[4] K. C. Das and I. Gutman, Some properties of the second Zagreb index, MATCH Commun. Math. Comput. Chem., 52 (2004), 103-112. 9
[5] S. Ediz, On ve-degree molecular topological properties of silicate and oxygen networks, Int. J. Comput. Sci. Math. 9(1) (2018) 1-12.
[6] S. Ediz, Predicting some physicochemical properties of octane isomers: A topological approach using evdegree and ve-degree Zagreb indices, Int. J. Sys. And Sci. Appl. Math. Vol. 2, No. 5, 2017, pp. 87-92.
[7] J. Gallian, Dynamic Survey DS6, Graph Labeling, Electronic J. Combin., DS6, (2007), 1-58.
[8] M. Ghorbani and M. A. Hosseinzadeh, A New Version of Zagreb Indices, Filomat, 21(1) (2012), 93-100.
[9] I. Gutman, Multiplicative Zagreb indices of trees, Bull. Soc. Math. Banja. Luka., 18 (2011), 17-23.
[10] I. Gutman, B. Ruscic, N. Trinajstic and C. F. Wilcox, Graph theory and molecular orbitals, XII. Acyclic polyenes, J. Chem. Phys., 62 (1975), 3399-3405.
[11] I. Gutman and N. Trinajstic, Graph theory and molecular orbitals. Total π-electron energy of Alternant hydrocarbons, Chem. Phys. Lett., 17 (1972), 535-538.
[12] F. Harary, Graph Theory, Addison-Wesley Publishing Co., Reading, Mass. Menlo Park, Calif. London, 1969.
[13] V. R. Kulli, B. Chaluvaraju and H. S. Boregowda, Some degree based connectivity indices of Kulli cycle windmill graph, South Asain J. Maths. 6(6) (2016), 263-268.
[14] P. Shiladhar, A. M. Naji and N. D.Soner, Leap Zagreb indices of Some wheel related Graphs, J. Comp. Math. Sci., 9(3) (2018), 221-231.
[15] P. Shiladhar, A. M. Naji and N. D.Soner, Computation of leap Zagreb indices of Some Windmill Graphs, Int. J. Math. And Appl., 6(2-B) (2018), 183-191.
[16] D. Vukicevic and A. Graovac, Note on the Comparison of the First and Second Normalized Zagreb Eccentricity Indices, Acta Chimica Slovenica, 57 (2010), 524-528.
[17] K. Xu and H. Hua, A unified approach to extremal multiplicative Zagreb indices for trees, unicyclicand bicyclic graphs, MATCH Commun. Math. Comput. Chem., 68 (2012), 241- 256

Citation :

Shiladhar Pawar, "On Vertex-Edge Eccentric Connectivity Index of Wheel related and Windmill Graphs," International Journal of Mathematics Trends and Technology (IJMTT), vol. 68, no. 4, pp. 43-51, 2022. Crossref, https://doi.org/10.14445/22315373/IJMTT-V68I4P508