Maximum Balaban index and sum-Balaban index of cacti

Abstract

The Balaban index and the sum-Balaban index of a graph [Formula: see text] are defined as [Formula: see text] and [Formula: see text], respectively, where [Formula: see text] is the distance sum of vertex [Formula: see text] in [Formula: see text], [Formula: see text] is the number of edges and [Formula: see text] is the cyclomatic number of [Formula: see text]. A connected graph [Formula: see text] is said to be a cactus if each of its blocks is either a cycle or an edge. In this paper, we determine the maximum values of Balaban index and sum-Balaban index and characterize the corresponding extremal graphs among all cacti with [Formula: see text] vertices and [Formula: see text] cycles.