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Let $G$ be a connected graph of order $n$.The Wiener index $W(G)$ of $G$ isthe sum of the distances between all unordered pairs of vertices of $G$. Inthis 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 oforder $n$ and minimum degree $\delta$ [M. Kouider, P. Winkler, Mean distanceand minimum degree. J. Graph Theory 25 no. 1 (1997)] can be improvedsignificantly 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 Wienerindex 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 determinea bound on the Wiener index of $C_4$-free graphs of given order, minimum andmaximum degree and show that it is, in some sense, best possible.

Wiener index in graphs with given minimum degree and maximum degree