Spectral Generalized Turán Problems for Expansion Hypergraphs

Abstract

Let $Q$ be an $r$-graph on $q$ vertices and let $\alpha>1$. The $(\alpha,Q)$-spectral radius $\lambda_{\alpha,Q}(\mathcal{H})$ of an $r$-graph $\mathcal{H}$ is defined as the maximum value of the $Q$-Lagrangian polynomial over all vectors with unit $\alpha$-norm. Given a family $\mathcal{F}$ of $r$-graphs, the ($\alpha, Q$)-spectral Turán number $\operatorname{spex}_\alpha(n, Q, \mathcal{F})$ of $\mathcal{F}$ is the maximum $(\alpha,Q)$ spectral radius $\lambda_{\alpha, Q}(\mathcal{H})$ among all $n$-vertex $\mathcal{F}$-free $r$-graphs $\mathcal{H}$. This spectral parameter provides a spectral counterpart of the generalized Turán problem and contains many classical spectral extremal problems as special cases.

In this paper, we first prove that for any $\alpha>1$ and every $2$-covered $r$-graph $Q$ with $q\le \ell$ vertices, the complete balanced $\ell$-partite $r$-graph $T_{\ell,n}^r$ uniquely maximizes $\lambda_{\alpha,Q}(\mathcal{H})$ among all $n$-vertex $\ell$-partite $r$-graphs. Based on this result and spectral stability theorems, we further determine the exact value of $\operatorname{spex}_\alpha(n,Q,H_F^r)$ for all sufficiently large $n$, where $H_F^r$ stands for the expansion of an edge-critical graph $F$ with chromatic number $\ell+1$. As corollaries, we obtain several Turán-type results for graphs and $r$-graphs, including spectral bounds for the clique spectral radius of $F$-free graphs and the $\alpha$-spectral radius of $H_{F}^r$-free $r$-graphs, where $F$ is an edge-critical graph.