On revised Szeged index of a class of unicyclic graphs

Abstract

Computing topological indices of graphs is a fundamental and classical topic. Let [Formula: see text] be a connected graph. The revised Szeged index [Formula: see text] is defined as [Formula: see text], where [Formula: see text] (respectively, [Formula: see text]) is the number of vertices whose distance to vertex [Formula: see text] (respectively, [Formula: see text]) is smaller than the distance to vertex [Formula: see text] (respectively, [Formula: see text]), and [Formula: see text] is the number of vertices equidistant from both ends of [Formula: see text]. In this paper, we determine the smallest revised Szeged index among all conjugated unicyclic graphs (i.e., unicyclic graphs with perfect matchings), and the corresponding extremal graphs are characterized.