Spectral Erdős--Gallai Theorems for the \(\mathcal A_α\)-Tensor of the \(s\)-Clique Hypergraph

Abstract

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.