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

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

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