Spectral Extremal Results for Hypergraphs

Yuan Hou,Joshua Cooper,An Chang

doi:10.37236/9018

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 Turán-type problems over linear $k$-uniform hypergraphs by using spectral methods, including a tight result on Berge $C_4$-free linear $3$-uniform hypergraphs.

Highlights

A hypergraph H = (V, E) consists of a vertex set V and an edge set E, where each edge is a nonempty subset of V
Given a family of graphs F, we say that a hypergraph H is Berge F-free if for every F ∈ F, the hypergraph H does not contain a Berge F as a subhypergraph
Our aim is to consider a spectral version of hypergraph Turan problems, i.e., spectral extremal hypergraph theory, which is the subset of extremal problems where invariants are based on the eigenvalues or eigenvectors of a hypergraph

Summary

Introduction

A hypergraph H = (V, E) consists of a vertex set V and an edge (hyperedge) set E, where each edge is a nonempty subset of V. A hypergraph is called k-uniform if each edge is a k-element subset of V. A 2-uniform hypergraph is called a graph. A hypergraph H is called linear if every two edges have at most one vertex in common. For a fixed k-uniform family F, the Turan number of F, denoted by exk(n, F), is the maximum number of edges of an F-free hypergraph on n vertices. Given a family of k-uniform linear hypergraphs F , the linear Turan number of F , denoted exlkin(n, F ), is the maximum number of edges in an F-free k-uniform linear hypergraph on n vertices

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