Coloring the normalized Laplacian for oriented hypergraphs

TL;DR

This paper explores spectral properties of the normalized Laplacian in oriented hypergraphs to establish bounds and relationships among independence, coloring, and clique parameters.

Contribution

It introduces spectral bounds and relationships for hypergraph parameters using the normalized Laplacian spectrum, extending spectral graph theory to hypergraphs.

Findings

Bounds for independence number using spectrum

A Sandwich Theorem relating clique, chromatic, and coloring numbers

Lower bounds for vector chromatic number based on eigenvalues

Abstract

The independence number, coloring number and related parameters are investigated in the setting of oriented hypergraphs using the spectrum of the normalized Laplace operator. For the independence number, both an inertia--like bound and a ratio--like bound are shown. A Sandwich Theorem involving the clique number, the vector chromatic number and the coloring number is proved, as well as a lower bound for the vector chromatic number in terms of the smallest and the largest eigenvalue of the normalized Laplacian. In addition, spectral partition numbers are studied in relation to the coloring number.