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arXiv · 2607.20383

The inverse problem for the Steiner-Wiener index of trees

Adrian Beker·Rudi Mrazović
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Abstract

For a connected graph $G$ and a set $S\subset V(G)$, the Steiner distance $d_G(S)$ is the minimum number of edges in a connected subgraph of $G$ containing $S$. The Steiner-Wiener $k$ index is defined by $\mathrm{SW}_k(G) = \sum_{S\subset V(G), |S|=k} d_G(S)$. We study the inverse problem for this invariant restricted to trees: for fixed $k$, which positive integers occur as $\mathrm{SW}_k(T)$ for a finite tree $T$? We prove that all sufficiently large positive integers occur as $\mathrm{SW}_k(T)$ for some finite tree $T$ if and only if $k$ is even. For odd $k$, we further show that the set of attainable values has asymptotic density of order $k^{-\delta}(\log k)^{-3/2}$, where $\delta$ is the Erd\H{o}s-Tenenbaum-Ford constant.

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Adrian Beker, Rudi Mrazović. 2026-07-22. The inverse problem for the Steiner-Wiener index of trees. https://arxiv.org/abs/2607.20383

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