On the Spectral Turán Problems for Bipartite Hypergraphs

Abstract

Given a graph $F$ and a positive integer $r\geq 2$, the $r$-expansion of $F$, denoted by $F^+$, is the $r$-graph obtained from $F$ by enlarging each edge of $F$ with $r$-$2$ new vertices disjoint from $V(F)$ such that distinct edges of $F$ are enlarged by distinct vertices. In this paper, we present some sharp bounds on the spectral radius of $K_{s,t}^+$-free linear $r$-graphs by establishing the connection between the spectral radius and the number of walks in uniform hypergraphs. For $t\geq 2$, we show that the spectral radius of a $K_{2,t}^+$-free $n$-vertex linear $r$-graph is at most $\frac{\sqrt{t-1}}{r-1}\sqrt{n}+O(1)$, which is close to being asymptotically optimal when $r=3$. Meanwhile, we prove that the spectral radius of a $K_{s,t}^+$-free $n$-vertex linear $r$-graph is $O(n^{1-\frac{1}{s}})$, where $t\geq s\geq 2$. The exponent of this upper bound is tight for $t>(s-1)!$ and $r=3$.