Spectral Erdős--Gallai Theorems for the ${\mathcal{A}}_{\alpha }$-Tensor of the $s$-Clique Hypergraph

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 ${\mathcal{A}}_{\alpha }$-tensor of the $s$-uniform clique hypergraph. For $0\le \alpha \le 1$ and $3\le s\le 2t-1$, we determine the maximum $\alpha$-$s$-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 $n\ge 2t-1$, it is attained by a clique of order $2t-1$ together with isolated vertices. At $\alpha =0$, these statements recover the known result for the $s$-clique spectral radius; at $\alpha =1$, they yield the corresponding statement for the maximum $s$-clique degree. For $s=t\ge 3$, we obtain the maximum for every $n\ge 2t-1$; at $\alpha =0$, this removes the requirement that $n$ be sufficiently large from the known result.