Spectral Turán-type problems for the $\alpha$-spectral radius of hypergraphs with degree stability

Abstract

An $r$-pattern $P$ is an ordered pair $P=([l],E)$, where $l$ is a positive integer and $E$ is a set of $r$-multisets with elements from $[l]$. An $r$-graph $H$ is said to be $P$-colorable if there is a homomorphism $\varphi$: $V(H)\to [l]$ such that $\{\varphi (v_1),\dots ,\varphi (v_r)\}\in E$ for every edge $\{v_1,\dots ,v_r\}\in E(H)$. Let $\mathrm{Col}(P)$ denote the family of all $P$-colorable $r$-graphs. This paper studies spectral extremal problems for $\alpha$-spectral radius of hypergraphs via analytic techniques. We first prove that for any $r$-pattern $P$, the hypergraph attaining the maximum $\alpha$-spectral radius in $\mathrm{Col}(P)$ is asymptotically regular. Specifically, we establish asymptotically tight lower bounds for the minimum component of the principal eigenvector and the minimum degree of the spectral extremal hypergraphs in $\mathrm{Col}(P)$. Building on this regularity, we further show that for any family $\mathcal{F}$ of $r$-graphs that is degree-stable with respect to $\mathrm{Col}(P)$, spectral Turán-type problems can be completely reduced to spectral extremal problems within $\mathrm{Col}(P)$. As an application, we determine the maximum $\alpha$-spectral radius ($\alpha \ge 1$) among all $n$-vertex $F^{(r)}$-free $r$-graphs, where $F^{(r)}$ is the $r$-expansion of the color-critical graph $F$. This provides a powerful reduction tool for handling spectral Turán-type problems in hypergraphs. Finally, leveraging the spectral method, we derive a corresponding edge Turán extremal result. More precisely, we show that if $\mathcal{F}$ is degree-stable with respect to $\mathrm{Col}(P)$, then every $\mathcal{F}$-free edge extremal hypergraph must be a $P$-colorable hypergraph.