For all n ≥ k ≥ 1, every maximal k-degenerate graph has Wiener index at least n^2 - (k+1)n + k(k+1)/2 and at most sum_{i=0}^{floor((n-2)/k)} C(n-ik,2), and for k-trees with n ≥ 2k+2 the upper bound is attained only by P_n^k.
An upper bound on the Wiener Index of a k-connected graph
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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 $H_{k,n}$.
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2019 1verdicts
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Wiener indices of maximal $k$-degenerate graphs
For all n ≥ k ≥ 1, every maximal k-degenerate graph has Wiener index at least n^2 - (k+1)n + k(k+1)/2 and at most sum_{i=0}^{floor((n-2)/k)} C(n-ik,2), and for k-trees with n ≥ 2k+2 the upper bound is attained only by P_n^k.