Hoffman's bound for hypergraphs

TL;DR

This paper extends Hoffman's spectral bound, originally for graphs, to weighted uniform hypergraphs for even r, providing a new tool for analyzing hypergraph chromatic numbers.

Contribution

It generalizes Hoffman's bound from graphs to weighted uniform hypergraphs for even r, broadening spectral graph theory applications.

Findings

Hoffman's inequality is extended to hypergraphs.

The bound applies to weighted uniform r-graphs for even r.

Provides a new spectral tool for hypergraph coloring analysis.

Abstract

One of the best-known results in spectral graph theory is the inequality of Hoffman \[ \chi\left( G\right) \geq1-\frac{\lambda\left( G\right) }{\lambda_{\min }\left( G\right) }, \] where $\chi\left( G\right) $ is the chromatic number of a graph $G$ and $\lambda\left( G\right) ,$ $\lambda_{\min}\left( G\right) $ are the largest and the smallest eigenvalues of its adjacency matrix. In this note Hoffman's inequality is extended to weighted uniform $r$-graphs for every even $r$.