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

A connected graph is said to be a cactus if each of its blocks is either a cycle or an edge. Let Cn be the set of all n-vertex cacti with circumference at least 4, and let Cn,k be the set of all n-vertex cacti containing exactly k⩾1 cycles where n⩾3k+1. In this paper, lower bounds on the difference between the (revised) Szeged index and Wiener index of graphs in Cn (resp. Cn,k) are proved. The minimum and the second minimum values on the difference between the Szeged index and Wiener index of graphs among Cn are 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 among Cn,k is also established. Along the way the corresponding extremal graphs are identified.