  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.
