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

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.