树的谱游走判定的一个严格匹配数阈值
A Sharp Matching-Number Threshold for Spectral-Walk Determination of Trees
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中文总结 AI 辅助
本文确定树的谱游走判定的匹配数阈值：匹配数≤4的树可由邻接谱与3≤k≤8的总游走序列唯一确定，匹配数为5时存在反例，证明基于有限核约化与代数重构。
中文摘要 AI 辅助
图的谱刻画是谱图理论的核心问题之一。本文研究树是否由其广义谱在树类中唯一确定的问题，采用邻接谱与总游走序列$W_k(G)=\boldsymbol{1}^{\textsf{T}}A(G)^k\boldsymbol{1}$结合的等价表述，确定该树级重构问题的精确匹配数阈值。若树$T$和$T'$的匹配数不超过4，且具有相同的邻接谱和总游走序列，则$T\boldsymbol{\ncong}T'$；在该范围内，仅需要求$3\boldsymbol{\nleq}k\boldsymbol{\nleq}8$时$W_k$相等即可。该界是严格的：对每个正整数$m$，本文构造了一对匹配数为5、具有相同邻接谱和相同总游走序列的非同构树。正结果的证明基于有限核约化与可能悬挂附接的代数重构。
英文摘要
The spectral characterization of graphs is a central problem in spectral graph theory. In this paper we study when a tree is determined, among trees, by its generalized spectrum. We use the equivalent formulation given by the adjacency spectrum together with the total-walk sequence $W_k(G)=\mathbf 1^{\mathsf T}A(G)^k\mathbf 1$. We determine the exact matching-number threshold for this tree-level reconstruction problem. If $T$ and $T'$ are trees with matching number at most 4 and have the same adjacency spectrum and the same total-walk sequence, then $T\cong T'$. Moreover, in this range it is enough to require equality of $W_k$ for $3\le k\le8$. The bound is sharp: for every positive integer $m$ we construct a pair of non-isomorphic trees with matching number 5 having the same adjacency spectrum and identical total-walk sequences. The proof of the positive result is based on a finite-core reduction and an algebraic reconstruction of the possible pendant attachments.
