The upper bound of the spectral radius for the hypergraphs without Berge-graphs

Abstract

The spectral analogue of the Turán type problem for hypergraphs is to determine the maximum spectral radius for the hypergraphs of order $n$ that do not contain a given hypergraph. For the hypergraphs among the set of the connected linear $3$-uniform hypergraphs on $n$ vertices without the Berge-$C_l$, we present two upper bounds for their spectral radius and $\alpha$-spectral radius, which are related to $n$,$l$ and $\alpha$, where $C_l$ is a cycle of length $l$ with $l⩾5$, $n⩾3$ and $0⩽\alpha <1$. Let $B_s$ be an $s$-book with $s⩾2$ and $K_{s,t}$ be a complete bipartite graph with two parts of size $s$ and $t$, respectively, where $s,t⩾1$. For the hypergraphs among the set of the connected linear $k$-uniform hypergraphs on $n$ vertices without the Berge-$\{B_s,K_{2,t}\}$, we derive two upper bounds for their spectral radius and $\alpha$-spectral radius, which depend on $n$, $k$, $s$, and $\alpha$, where $n$,$k⩾3$,$s⩾2$,$1⩽t⩽\frac{1}{2}(6k^2-15k+10)(s-1)+1$, and $0⩽\alpha <1$.

16 pages