The (revised) Szeged index and the Wiener index of a nonbipartite graph

TL;DR

This paper confirms two conjectures relating the Szeged and revised Szeged indices to the Wiener index in nonbipartite graphs, providing proofs and characterizations of extremal graphs that attain the bounds.

Contribution

The paper proves two conjectures about bounds on the differences between Szeged indices and the Wiener index in nonbipartite graphs, and characterizes the extremal graphs.

Findings

Proved that Sz(G)-W(G)  2n-5 for certain nonbipartite graphs.

Confirmed that Sz^*(G)-W(G)  (n^2+4n-6)/4 for specific nonbipartite graphs.

Characterized graphs that achieve the lower bounds of these differences.

Abstract

Hansen et. al. used the computer programm AutoGraphiX to study the differences between the Szeged index $Sz(G)$ and the Wiener index $W(G)$, and between the revised Szeged index $Sz^*(G)$ and the Wiener index for a connected graph $G$. They conjectured that for a connected nonbipartite graph $G$ with $n \geq 5$ vertices and girth $g \geq 5,$ $ Sz(G)-W(G) \geq 2n-5. $ Moreover, the bound is best possible as shown by the graph composed of a cycle on 5 vertices, $C_5$, and a tree $T$ on $n-4$ vertices sharing a single vertex. They also conjectured that for a connected nonbipartite graph $G$ with $n \geq 4$ vertices, $ Sz^*(G)-W(G) \geq \frac{n^2+4n-6}{4}. $ Moreover, the bound is best possible as shown by the graph composed of a cycle on 3 vertices, $C_3$, and a tree $T$ on $n-3$ vertices sharing a single vertex. In this paper, we not only give confirmative proofs to these two conjectures but also characterize those graphs that achieve the two lower bounds.