An upper bound on the Wiener Index of a k-connected graph

Abstract: The Wiener index of a connected graph is the summation of all distances between unordered pairs of vertices of the graph. In this paper, we give an upper bound on the Wiener index of a $k$-connected graph $G$ of order $n$ for integers $n-1>k \ge 1$: [W(G) \le \frac{1}{4} n \lfloor \frac{n+k-2}{k} \rfloor (2n+k-2-k\lfloor \frac{n+k-2}{k} \rfloor).] Moreover, we show that this upper bound is sharp when $k \ge 2$ is even, and can be obtained by the Wiener index of Harary graph $H_{k,n}$.