谱 Erdős--Gallai 定理：关于 $s$-团超图的 $\mathcal A_\alpha$-张量
Spectral Erdős--Gallai Theorems for the \(\mathcal A_α\)-Tensor of the \(s\)-Clique Hypergraph
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中文总结 AI 辅助
本文研究 $s$-团超图的 $\mathcal A_\alpha$-张量谱半径，确定了无 $t$ 匹配的图中 $\alpha$-$s$-团谱半径的最大值，并推广了已知结果，覆盖了 $\alpha=0$ 和 $\alpha=1$ 的情形。
中文摘要 AI 辅助
Erdős--Gallai 定理确定了在匹配数有界时图中边数的最大值；其团计数推广将边替换为 $s$-团，而最近已建立了基于 $s$-团张量的谱类比。我们研究 $s$-均匀团超图的相应 $\mathcal A_\alpha$-张量。对于 $0\leq\alpha\leq1$ 和 $3\le s\le2t-1$，我们确定了在包含 $t$ 条边匹配的 $n$ 顶点图中 $\alpha$-$s$-团谱半径的最大值：当 $3\le s\le t$ 且 $n$ 足够大时，最大值由阶为 $t-1$ 的团与独立集的并（join）取得；当 $t<s\le 2t-1$ 且 $n\ge2t-1$ 时，最大值由阶为 $2t-1$ 的团与孤立顶点取得。在 $\alpha=0$ 时，这些陈述恢复了已知的 $s$-团谱半径结果；在 $\alpha=1$ 时，它们给出了最大 $s$-团度的相应陈述。对于 $s=t\ge3$，我们得到了所有 $n\ge2t-1$ 时的最大值；在 $\alpha=0$ 时，这从已知结果中移除了 $n$ 足够大的要求。
英文摘要
The Erdős--Gallai theorem determines the maximum number of edges in a graph with bounded matching number; its clique-counting extension replaces edges by $s$-cliques, and a spectral analogue in terms of the $s$-clique tensor has recently been established. We study the corresponding $\mathcal A_α$-tensor of the $s$-uniform clique hypergraph. For $0\leqα\leq1$ and $3\le s\le2t-1$, we determine the maximum $α$-$s$-clique spectral radius among $n$-vertex graphs containing no matching of $t$ edges: when $3\le s\le t$ and $n$ is sufficiently large, the maximum is attained by the join of a clique of order $t-1$ and an independent set, and when $t<s\le 2t-1$ and $n\ge2t-1$, it is attained by a clique of order $2t-1$ together with isolated vertices. At $α=0$, these statements recover the known result for the $s$-clique spectral radius; at $α=1$, they yield the corresponding statement for the maximum $s$-clique degree. For $s=t\ge3$, we obtain the maximum for every $n\ge2t-1$; at $α=0$, this removes the requirement that $n$ be sufficiently large from the known result.
发表机构
- Tongji University（同济大学）
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