Wiener index in graphs with given minimum degree and maximum degree

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 (n/delta+1 + 2) ((2)(n)) on the Wiener index of a graph of order n and minimum degree d [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) <= ((n-Delta+delta)(2)) n+2 Delta/delta+1 + 2n(n - 1) on the Wiener index W (G) 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.