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

TL;DR

This work develops a comprehensive framework for spectral extremal problems in hypergraphs via the $(p,Q)$-spectral radius $\lambda^{(p)}(Q,H)$ and its density $\pi(Q,\mathcal{P})$ for hereditary families $\mathcal{P}$. It proves the existence of the spectral limit $\lambda^{(p)}(Q,\mathcal{P})$ and shows that for $p>1$, $\lambda^{(p)}(Q,\mathcal{P})=\pi(Q,\mathcal{P})$, extending prior density–spectral radius connections. The paper then establishes a spectral stability principle and derives two major results: a spectral Erdős pentagon theorem, where the balanced blow-up of $C_5$ is extremal among triangle-free graphs for large $n$, and a spectral analogue for edge-critical graphs, where the Turán graph $T^{(s)}_{\ell}(n)$ maximizes $\lambda^{(p)}$ among $F^{(s)}$-free graphs for large $n$ (with $p>1$). These results extend classical Turán-type theorems into the spectral domain, using blow-up techniques, $Q$-flatness concepts, and stability arguments to connect density and spectral extremals in 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 $p\geq1$, the $(p,Q)$-spectral radius of $H$ is defined as $$λ^{(p)}(Q,H):=\max_{\|\mathbf{x}\|_{p}=1}s!\sum_{\{i_{1},\ldots,i_{s}\}\in \binom{[n]}{s}}\mathcal{N}(Q,H[\{i_{1},\ldots,i_{s}\}])x_{i_{1}}\cdots x_{i_{s}}.$$ %generalizing the concept of the $p$-spectral radius introduced by %Keevash, Lenz, and Mubayi \cite{KLM2014}. In this paper, we present a systematically investigation of the parameter $λ^{(p)}(Q,H)$. 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 $$π(Q,\mathcal{P})=λ^{(p)}(Q,\mathcal{P}).$$ 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 $p\geq1$ and sufficiently large $n$, the balanced blow-up of $C_{5}$ maximizes $λ^{(p)}(C_{5},H)$ among all $n$-vertex triangle-free graphs $H$, thereby improving a result of Liu \cite{Liu2025}. Furthermore, we show that for $p\geq1$ and sufficiently large $n$, the $l$-partite Turán graph $T_{l}(n)$ attains the maximum $λ^{(p)}(K_{s},H)$ among all $n$-vertex F-free graphs $H$, where $F$ is an edge-critical graph with $χ(F)=l+1$. This provides a spectral analogue of a theorem due to Ma and Qiu \cite{MQ2020}.

Theorems & Definitions (45)

• Proposition 2.1
• proof
• Lemma 2.2
• proof
• Proposition 2.3
• Theorem 3.2
• proof
• Theorem 3.3
• Lemma 3.4
• proof