Spectral Tur\'{a}n results on triangle-free graphs and hypergraphs

Abstract

One of the most classical spectral Tur\'{a}n problems is determining the maximum spectral radius of triangle-free graphs and hypergraphs. As the hypergraph analog of triangles, $Fan^{k}$ is a $k$-uniform linear hypergraph with $k$ edges
$f_{1}, \cdots, f_{k}$ which pairwise intersect in a common vertex $v$, and an additional edge $g$
which intersects all $f_{i}$ in a vertex different from $v$.
Let $K_{s, t}^{-}$ be the graph obtained from a complete bipartite graph $K_{s,t}$ by deleting an edge.
Motivated by the classic theorems on triangle-free graphs due to Erd\H{o}s
and Nosal, and by the Tur\'{a}n number of $Fan^{k}$ on $k$-uniform linear hypergraphs determined by F\"{u}redi and Gy\'{a}rf\'{a}s respectively, we prove that $\rho(G)\leq \rho(K_{\left \lfloor \frac{n}{2} \right \rfloor , \left \lceil \frac{n}{2} \right \rceil }^{-})$ where $\rho(G)$ is the spectral radius of a connected triangle-free graph $G$ with order $n$ and diameter 3, and $spex_{k}^{lin}(m, Fan^{k})=\sqrt{m}$ where $spex_{k}^{lin}(m, Fan^{k})$ denotes the maximum spectral radius of $Fan^{k}$-free linear $k$-uniform hypergraphs with size $m$.