Spectral characterizations of local structures of graphs and hypergraphs

TL;DR

This paper explores how spectral properties of graphs and hypergraphs relate to their local structures, establishing conditions for subgraph occurrences and bounds on local coloring parameters using tensor eigenvalues.

Contribution

It introduces new spectral bounds on local vector chromatic numbers and links spectral radius ratios to the presence of specific local substructures.

Findings

Large spectral radius ratios imply the existence of certain local subgraphs.

Spectral bounds on local vector chromatic number are derived using tensor eigenvalues.

The work connects spectral properties with local structural features of graphs and hypergraphs.

Abstract

In this paper, we give the relationship between spectral radius and local structures of graphs and hypergraphs. Our work shows that certain local subgraphs (subhypergraphs) must occur when the spectral radius ratio is large. We also give spectral bounds on the local vector chromatic number in terms of tensor eigenvalues of graphs.