On the difference between the (revised) Szeged index and the Wiener index of cacti

摘要

A connected graph is said to be a cactus if each of its blocks is either a cycle or an edge. LetCnbe the set of alln-vertex cacti with circumference at least 4, and letCn,kbe the set of alln-vertex cacti containing exactlyk?1cycles wheren?3k+1. In this paper, lower bounds on the difference between the (revised) Szeged index and Wiener index of graphs inCn(resp.Cn,k) are proved. The minimum and the second minimum values on the difference between the Szeged index and Wiener index of graphs amongCnare determined. The bound on the minimum value is strengthened in the bipartite case. A lower bound on the difference between the revised Szeged index and Wiener index of graphs amongCn,kis also established. Along the way the corresponding extremal graphs are identified.