The Atom Bond Connectivity Index of Certain Graphs

Abstract

The atom bond connectivity (ABC) index provides a good model for the stability of linear and branched alkanes as well as the strain energy of cycloalkanes, which is defined as ABC = uv∈E(G) du+dv −2 dudv , we denotes du the degree of vertex u in G. In this paper, we study the general formula for ABC index of certain graphs and vertex gluing of graphs.

RETRACTED ARTICLE: Topological analysis of carbon and boron nitride nanotubes

“…In 21 , Ghorbani with co-authors gave the closed formulae to compute the fourth version of atom-bond connectivity index (ABC 4 ). The relation to evaluate the ABC 4 index in terms of sum of degrees of neighboring vertices is; For more information see [23][24][25][26][27] .…”

On Harmonic Index of Some Special Graphs with Certain Vertex Gluing Graphs

“…"A K 4 -homeomorphism graph is the graph obtained when the six edges of a complete graph with four vertices (K 4 ) are subdivided into (e 1 , e 2 , e 3 , e 4 , e 5 , e 6 ) segments, respectively [8] " see " Fig. 1,". "…”

“…So K n,m has order (n+m) and size (nm). A tree K 1,m is also called a star [8]" see " Fig "A (r)bridge graph is a graph consisting of s paths joining two vertices, which is denoted by Q(a 1 ,……..,a r ), where a 1 ,…….,a r , are the lengths of r paths [8]" see " Fig "A web graph Web(n,m) is the graph obtained from the Cartesian product of the cycle C n and the path P m [8]" see " Fig. 4,".…”

“…10. (r+1 (vertex gluing of graph ) [8]", see " Fig. 11," Suppose that n, m, h and t are positive integers, then, the v 1gluing of a complete bipartite graph H -index…”

“…Khomeomorphism be a graph obtained from two deferent K 4homeomorphism graphs K 4 (e 1 , e 2 , e 3 , e 4 , e 5 , e 6 ) and K 4 (e 7 , e 8 , e 9 , e 10 , e 11 , e 12 ), with common one vertex v 1 (vertex gluing of graph) [8]", see " Fig. 12,".…”