A relation between the multiplicity of the second eigenvalue of a graph Laplacian, Courant's nodal line theorem and the substantial dimension of tight polyhedral surfaces

TL;DR

This paper explores the relationship between the second eigenvalue's multiplicity of a graph Laplacian, tight mappings, and Courant's theorem, revealing bounds related to graph properties like chromatic number and genus.

Contribution

It establishes a connection between eigenvalue multiplicity, tight graph mappings, and topological bounds, providing new insights into spectral graph theory.

Findings

The m-dimensional eigenspace of λ2 is tight for certain graphs.

Tightness of the mapping bounds the eigenvalue multiplicity.

Links eigenvalue multiplicity to graph chromatic number and genus.

Abstract

This note discusses a relation between the multiplicity m of the second eigenvalue {\lambda}2 of a Laplacian on a graph G, tight mappings of G and a discrete analogue of Courant's nodal line theorem. For a certain class of graphs, we show that the m-dimensional eigenspace of {\lambda}2 is tight and thus defines a tight mapping of G into an m-dimensional Euclidean space. The tightness of the mapping is shown to set Colin de Verdi\`ere's upper bound on the maximal {\lambda}2-multiplicity, where chr({\gamma}(G)) is the chromatic number and {\gamma}(G) is the genus of G.