Spectral Radius Conditions for 3-Uniform Intersecting Families

Let $M_k$ denote a matching of size $k$. The classical Erd\H{o}s matching conjecture asks for the maximum number of edges of an intersecting $r$-graph without $M_k$. The csae for $k=2$, which is known as intersecting $r$-graph, is established by Erd\H{o}s, Ko and Rado. Hilton and Milner further determine the maximum number of edges of a non-trivial intersecting $r$-graph, where the intersecting $r$-graph $H$ is called non-trivial if $\cap_{e\in E(H)}e=\emptyset$. In this paper, we investigate the spectral analogues of the hpergraph matching problems and intersecting family problems. More precisely, for sufficiently large $n$, we determine respectively the maximum spectral radius of $M_{k+1}$-free and non-trivial intersecting $3$-graphs on $n$ vertices, and characterize the extremal hypergraphs.