Nodal domain counts and the chromatic number of graphs

Abstract

A function f, defined on the vertices of a graph G, induces nodal domains on the graph. Nodal domains of discrete and metric graphs are of growing interest among physicists and mathematicians. In this paper, several results regarding the nodal domain counts of discrete graphs are derived. One such result is a global upper bound for the number of nodal domains of G, in terms of its chromatic number. Another result is a criterion of resolution of (Laplacian) isospectral graphs via their nodal counts. Several additional results are also shown. © 2007 IOP Publishing Ltd.