Nowhere-harmonic colorings of graphs

Beck, Matthias; Braun, Benjamin

doi:10.1090/S0002-9939-2011-10879-7

Nowhere-harmonic colorings of graphs
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by Matthias Beck and Benjamin Braun PDF

Proc. Amer. Math. Soc. 140 (2012), 47-63 Request permission

Abstract:

Proper vertex colorings of a graph are related to its boundary map $\partial _1$, also called its signed vertex-edge incidence matrix. The vertex Laplacian of a graph, $L=\partial _1 \partial _1^t$, a natural extension of the boundary map, leads us to introduce nowhere-harmonic colorings and analogues of the chromatic polynomial and Stanley’s theorem relating negative evaluations of the chromatic polynomial to acyclic orientations. Further, we discuss several examples demonstrating that nowhere-harmonic colorings are more complicated from an enumerative perspective than proper colorings.

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