Bounds on the Spectral Radii of Berge $C_5$-Free Linear $r$-Graphs

Abstract

An $r$-uniform hypergraph (or $r$-graph) is called linear if any two edges intersect in at most one vertex. For a graph $F=\bigl(V(F),E(F)\bigr)$ and a hypergraph $\mathcal{B}=\bigl(V(\mathcal{B}),E(\mathcal{B})\bigr)$, $\mathcal{B}$ is called a Berge $F$ if there exists a bijection $\phi:E(F)\to E(\mathcal{B})$ such that $e\subseteq \phi(e)$ for every $e\in E(F)$. A hypergraph $H$ is Berge $F$-free if it contains no Berge $F$ as a subhypergraph. Hou et al. [Electron. J. Combin. 28 (2021)] derived a upper bound for the spectral radius of Berge $C_4$-free linear $r$-graphs. In this paper, we establish upper bounds for the spectral radius of Berge $C_5$-free linear $r$-graphs for $r=3$ and $r\ge 4$. Moreover, for $r>4$, we propose a candidate extremal structure for the hypergraph with maximum spectral radius among all Berge $C_5$-free linear $r$-graphs.