树的Steiner-Wiener指标的逆问题
The inverse problem for the Steiner-Wiener index of trees
浏览论文内容
中文总结 AI 辅助
研究树的Steiner-Wiener \(k\)指标的逆问题，即固定\(k\)时，哪些正整数可作为有限树\(T\)的\(\mathrm{SW}_k(T)\)。证明\(k\)为偶数时所有足够大正整数都可出现，奇数\(k\)时可达到值集合有特定渐近密度。
中文摘要 AI 辅助
对于连通图\(G\)和集合\(S\subset V(G)\)，Steiner距离\(d_G(S)\)是\(G\)中包含\(S\)的连通子图的最小边数。Steiner-Wiener \(k\)指标定义为\(\mathrm{SW}_k(G)=\sum_{S\subset V(G), |S|=k} d_G(S)\)。我们研究此不变量在树中的逆问题：对于固定的\(k\)，哪些正整数会作为有限树\(T\)的\(\mathrm{SW}_k(T)\)出现？我们证明，当且仅当\(k\)为偶数时，所有足够大的正整数都会作为某个有限树\(T\)的\(\mathrm{SW}_k(T)\)出现。对于奇数\(k\)，我们进一步表明可达到值的集合具有阶为\(k^{-\delta}(\log k)^{-3/2}\)的渐近密度，其中\(\delta\)是厄尔多斯 - 特南鲍姆 - 福特常数。
英文摘要
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 $k^{-δ}(\log k)^{-3/2}$, where $δ$ is the Erdős-Tenenbaum-Ford constant.
