On extremal cacti with respect to the revised Szeged index

Abstract

The revised Szeged index of a graph G is defined as Sz∗(G)=∑e=uv∈E(nu(e)+n0(e)2)(nv(e)+n0(e)2), where nu(e) and nv(e) are, respectively, the number of vertices of G lying closer to vertex u than to vertex v and the number of vertices of G lying closer to vertex v than to vertex u, and n0(e) is the number of vertices equidistant to u and v. A cactus is a graph in which any two cycles have at most one common vertex. Let C(n,k) denote the class of all cacti with n vertices and k cycles. In this paper, sharp lower bound on revised Szeged index of graph G in C(n,k) is established and the corresponding extremal graph is determined. Furthermore, the graph G in C(n,k) with the second minimal revised Szeged index is identified as well.