| Summary: | Let $G$ be a connected graph of order $n$.The Wiener index $W(G)$ of $G$ is
the sum of the distances between all unordered pairs of vertices of $G$. In
this paper we show that the well-known upper bound $\big(
\frac{n}{\delta+1}+2\big) {n \choose 2}$ on the Wiener index of a graph of
order $n$ and minimum degree $\delta$ [M. Kouider, P. Winkler, Mean distance
and minimum degree. J. Graph Theory 25 no. 1 (1997)] can be improved
significantly if the graph contains also a vertex of large degree.
Specifically, we give the asymptotically sharp bound $W(G) \leq
{n-\Delta+\delta \choose 2} \frac{n+2\Delta}{\delta+1}+ 2n(n-1)$ on the Wiener
index of a graph $G$ of order $n$, minimum degree $\delta$ and maximum degree
$\Delta$. We prove a similar result for triangle-free graphs, and we determine
a bound on the Wiener index of $C_4$-free graphs of given order, minimum and
maximum degree and show that it is, in some sense, best possible.
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