On the entire ABC index of graphs

Abstract

Topological indices are graph invariants computed usually by means of the distances or vertex degrees of molecular graphs. In chemical graph theory, topological indices have been successfully used in describing the structure and also predicting certain physico-chemical properties of chemical compounds. Atom-bond connectivity (ABC) index has been applied to the study of the stability of alkanes and to the strain the energy of cycloalkanes. The atom bond connectivity (ABC) index of a graph G is defined as ABC(G)= ΣuvϵE(G)√deg(u)+deg(v)-2/deg(u)deg(v) where E{G) denotes the set of edges of G and deg{u) and deg(v) are the degrees of the vertices u and v, respectively. Recently, for several applications, the entire versions of the topological indices are defined and studied. In this versions of the topological indices, not only the vertices or the edges are considered in calculations. Both of them are used instead. In this research, we introduce and study the entire atom bond connectivity index of a graph. Exact values of this index for some families of graphs are obtained and some important properties of this new index are established.