AI 中文总结
本文研究了具有给定匹配数的连通图的谱半径下界，并给出了极值图的刻画，同时证明了谱半径与匹配数之和的渐进行为为n的1/3次方。
AI 中文摘要
设\(\mathscr{G}_{n,k}\)表示所有阶为\(n\)且匹配数为\(k\)的连通图的族。Liu、Lou和Trevisan（《线性代数及其应用》，2026）提出如下问题：确定\(\mathscr{G}_{n,k}\)中谱最小的图。本文证明，对于每个\(G\in\mathscr{G}_{n,k}\)，\(\rho(G)\geq\sqrt{\frac{n + 2k - 3}{k}}\)，并完全刻画了\(k\mid(n - 3)\)时的极图。作为应用，对于\(k\geq2\)，建立了\(\rho(G)+k\geq3\sqrt[3]{n/4}\)，确定了\(\rho + k\)的渐近阶为\(\Theta(n^{{1}/{3}})\)，严格小于被否定的Aouchiche - Hansen猜想所暗示的\(\Theta(\sqrt{n})\)阶。
英文摘要
Let $\mathscr{G}_{n,k}$ denote the family of all connected graphs of order $n$ with matching number $k$. Liu, Lou, and Trevisan~(Linear Algebra Appl., 2026) posed the following problem: Determine the spectrally minimal graphs in $\mathscr{G}_{n,k}$. In this paper we prove that for every graph $G \in \mathscr{G}_{n,k}$, $ ρ(G) \ge \sqrt{\frac{n + 2k - 3}{k}}, $ and we completely characterize the extremal graphs when $k \mid (n-3)$. As applications, we establish $ρ(G) + k \ge 3\sqrt[3]{n/4}$ for $k \ge 2$, settling the asymptotic order of $ρ+ k$ as $Θ(n^{1/3})$ -- strictly smaller than the $Θ(\sqrt{n})$ order suggested by the disproved Aouchiche--Hansen conjecture.
Comments11 pages, 3 figures
