Spectral Radius on Linear $r$-Graphs without Expanded $K_{r+1}$

Abstract

An $r$-uniform hypergraph is linear if every two edges intersect in at most one vertex. Let $K_{r+1}$ be a complete graph with $r+1$ vertices. The $r$-uniform hypergraph $K_{r+1}^+$ is obtained from $K_{r+1}$ by enlarging each edge of $K_{r+1}$ with $r-2$ new vertices disjoint from $V(K_{r+1})$ such that distinct edges of $K_{r+1}$ are enlarged by distinct vertices. Let $H$ be a $K_{r+1}^+$-free linear $r$-uniform hypergraph with $n$ vertices. In this paper, we prove that when $n$ is sufficiently large, the spectral radius $\rho (H)$ of the adjacency tensor of $H$ is no more than $\frac{n}{r}$, i.e., $\rho (H)\leq \frac{n}{r}$, with equality if and only if $r|n$ and $H$ is a transversal design, where the transversal design is the balanced $r$-partite $r$-uniform hypergraph such that each pair of vertices from distinct parts is contained in one hyperedge exactly. An immediate corollary of this result is that $ex_r^{lin}(n,K_{r+1}^+)= \frac{n^2}{r^2}$ for sufficiently large $n$ and $r|n$, where $ex_r^{lin}(n,K_{r+1}^+)$ is the maximum number of edges of an $n$-vertex $K_{r+1}^+$-free linear $r$-uniform hypergraph, i.e., the linear Turán number of $K_{r+1}^+$.