Colouring the normalized Laplacian

TL;DR

This paper derives eigenvalue bounds for the chromatic number using the normalized Laplacian, introduces new expansion parameters related to graph coloring, and explores their equality conditions with applications to specific graph families.

Contribution

It provides new eigenvalue bounds for graph coloring, characterizes equality cases, and introduces generalized expansion parameters related to the Cheeger constant.

Findings

Eigenvalue bounds for chromatic number derived from normalized Laplacian.

Characterization of equality cases in eigenvalue bounds.

Introduction of new expansion parameters generalizing the Cheeger constant.

Abstract

We apply Cauchy's interlacing theorem to derive some eigenvalue bounds to the chromatic number using the normalized Laplacian matrix, including a combinatorial characterization of when equality occurs. Further, we introduce some new expansion type of parameters which generalize the Cheeger constant of a graph, and relate them to the colourings which meet our eigenvalue bound with equality. Finally, we exhibit a family of examples, which include the graphs that appear in the statement of the Erd\H{o}s-Faber-Lov\'asz conjecture.