On revised Szeged index of a class of unicyclic graphs

Computing topological indices of graphs is a fundamental and classical topic. Let G be a connected graph. The revised Szeged index S-z*(G) is defined as S-z*(G) = Sigma(e=uv)(is an element of E)((G))(n(u)(e vertical bar G) + n(0)(e vertical bar G)/2) (n(v) (e vertical bar G) + n(0) (e vertical bar G)/2), where n(u) (e vertical bar G) (respectively, n(v) (e vertical bar G) is the number of vertices whose distance to vertex u (respectively, v) is smaller than the distance to vertex v (respectively, u), and n(0)(e vertical bar G) is the number of vertices equidistant from both ends of e. 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.