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

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 $S{z}^{\ast }(G)$ and the Wiener index for a connected graph $G$. They conjectured that for a connected nonbipartite graph $G$ with $n \geq 5$verticesandgirth$g \geq 5,$ $Sz(G)-W(G)\ge 2n-5.$ 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 $n\ge 4$ vertices, $S{z}^{\ast }(G)-W(G)\ge \frac{n^2+4n-6}{4}\mathrm{.}$ 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.