Spectral bounds for non-uniform hypergraphs using weighted clique expansion

TL;DR

This paper extends spectral bounds from uniform to non-uniform hypergraphs using weighted clique expansion, providing new eigenvalue bounds and connectivity insights.

Contribution

It generalizes spectral results for uniform hypergraphs to non-uniform cases and establishes bounds on eigenvalues and connectivity parameters.

Findings

Bound on the largest eigenvalue related to average degree

Inequality on boundary of vertex sets involving eigenvalues

Bounds on hypergraph connectivity parameters

Abstract

Hypergraphs are an invaluable tool to understand many hidden patterns in large data sets. Among many ways to represent hypergraph, one useful representation is that of weighted clique expansion. In this paper, we consider this representation for non-uniform hypergraphs. We generalize the spectral results for uniform hypergraphs to non-uniform hypergraphs and show that they extend in a natural way. We provide a bound on the largest eigenvalue with respect to the average degree of neighbours of a vertex in a graph. We also prove an inequality on the boundary of a vertex set in terms of the largest and second smallest eigenvalue and use it to obtain bounds on some connectivity parameters of the hypergraph.