On the difference between the Szeged and Wiener index

TL;DR

This paper proves a conjecture relating the difference between Szeged and Wiener indices in 2-connected non-complete graphs, characterizes extremal graphs, and extends results to bipartite and girth conditions.

Contribution

It establishes a lower bound for the Szeged-Wiener index difference in 2-connected graphs and characterizes the extremal cases, also strengthening related known results.

Findings

Proved the conjecture: Sz(G)-W(G) ≥ 2n-6 for 2-connected non-complete graphs.

Characterized the extremal graphs achieving equality.

Extended results to bipartite graphs and graphs with girth at least five.

Abstract

We prove a conjecture of Nadjafi-Arani, Khodashenas and Ashrafi on the difference between the Szeged and Wiener index of a graph. Namely, if $G$ is a 2-connected non-complete graph on $n$ vertices, then $Sz(G)-W(G)\ge 2n-6$. Furthermore, the equality is obtained if and only if $G$ is the complete graph $K_{n-1}$ with an extra vertex attached to either $2$ or $n-2$ vertices of $K_{n-1}$. We apply our method to strengthen some known results on the difference between the Szeged and Wiener index of bipartite graphs, graphs of girth at least five, and the difference between the revised Szeged and Wiener index. We also propose a stronger version of the aforementioned conjecture.