Spectral extremal problems for the $(p,Q)$-spectral radius of hypergraphs

Abstract

Let $Q$ be an $s$-vertex $r$-uniform hypergraph, and let $H$ be an $n$-vertex $r$-uniform hypergraph. Denote by $\mathcal{N}(Q,H)$ the number of isomorphic copies of $Q$ in $H$. For a hereditary family $\mathcal{P}$ of $r$-uniform hypergraphs, define $$π(Q,\mathcal{P}):=\lim\limits_{n\to \infty}\binom{n}{s}^{-1}\max\{\mathcal{N}(Q,H): H\in \mathcal{P}~~\mbox{and}~~|V(H)|=n\}.$$ For $p\ge 1$, the $(p,Q)$-spectral radius of $H$ is defined as $$\hat{I}{»}^{(p)}(Q,H):=\underset{\Vert \mathbf{x}{\Vert }_p=1}{\max }s!\sum _{\{i_1,\dots ,i_s\}\in \left(\genfrac{}{}{0ex}{}{[n]}{s}\right)}\mathcal{N}(Q,H[\{i_1,\dots ,i_s\}])x_{i_1}\cdots x_{i_s}.$$ In this paper, we present a systematically investigation of the parameter $\hat{I}{»}^{(p)}(Q,H)$. First, we prove that the limit $$λ^{(p)}(Q,\mathcal{P}):=\lim\limits_{n\to \infty}n^{s/p-s}\max\{λ^{(p)}(Q,H): H\in \mathcal{P}~~\mbox{and}~~|V(H)|=n\}$$ exists, and for $p>1$, it satisfies $$\ddot{I}€(Q,\mathcal{P})=\hat{I}{»}^{(p)}(Q,\mathcal{P}).$$ Second, we study spectral generalized Turán problems. Specifically, we establish a spectral stability result and apply it to derive a spectral version of the Erdős Pentagon Problem: for $p\ge 1$ and sufficiently large $n$, the balanced blow-up of $C_5$ maximizes $\hat{I}{»}^{(p)}(C_5,H)$ among all $n$-vertex triangle-free graphs $H$, thereby improving a result of Liu \cite{Liu2025}. Furthermore, we show that for $p\ge 1$ and sufficiently large $n$, the $l$-partite Turán graph $T_l(n)$ attains the maximum $\hat{I}{»}^{(p)}(K_s,H)$ among all $n$-vertex F-free graphs $H$, where $F$ is an edge-critical graph with $\ddot{I}‡(F)=l+1$. This provides a spectral analogue of a theorem due to Ma and Qiu \cite{MQ2020}.