The extremal $p$-spectral radius of Berge-hypergraphs

Abstract

Let $G$ be a graph. We say that a hypergraph $H$ is a Berge-$G$ if there is a bijection $\phi: E(G)\to E(H)$ such that $e\subseteq \phi(e)$ for all $e\in E(G)$. For any $r$-uniform hypergraph $H$ and a real number $p\geq 1$, the $p$-spectral radius $\lambda^{(p)}(H)$ of $H$ is defined as \[ \lambda^{(p)}(H):=\max_{{\bf x}\in\mathbb{R}^n,\,\|{\bf x}\|_p=1} r\sum_{\{i_1,i_2,\ldots,i_r\}\in E(H)} x_{i_1}x_{i_2}\cdots x_{i_r}. \] In this paper, we study the $p$-spectral radius of Berge-$G$ hypergraphs. We determine the $3$-uniform hypergraphs with maximum $p$-spectral radius for $p\geq 1$ among Berge-$G$ hypergraphs when $G$ is a path, a cycle or a star.