Spectral Extremal Results for Hypergraphs

TL;DR

This paper explores the relationship between spectral properties and structural features of linear hypergraphs, providing spectral versions of Turán-type problems and a tight result for Berge C4-free hypergraphs.

Contribution

It introduces spectral methods to analyze Turán-type extremal problems in linear hypergraphs, including new bounds and a specific tight result for Berge C4-free hypergraphs.

Findings

Spectral bounds for Berge F-free hypergraphs

A tight spectral Turán-type result for Berge C4-free linear hypergraphs

Connections established between spectral radius and hypergraph structure

Abstract

Let $F$ be a graph. A hypergraph is called Berge $F$ if it can be obtained by replacing each edge in $F$ by a hyperedge containing it. Given a family of graphs $\mathcal{F}$, we say that a hypergraph $H$ is Berge $\mathcal{F}$-free if for every $F \in \mathcal{F}$, the hypergraph $H$ does not contain a Berge $F$ as a subhypergraph. In this paper we investigate the connections between spectral radius of the adjacency tensor and structural properties of a linear hypergraph. In particular, we obtain a spectral version of Tur\'{a}n-type problems over linear $k$-uniform hypergraphs by using spectral methods, including a tight result on Berge $C_4$-free linear $3$-uniform hypergraphs.