Nowhere-harmonic colorings of graphs

Proper vertex colorings of a graph are related to its boundary map ∂₁, also called its signed vertex-edge incidence matrix. The vertex Laplacian of a graph, L = ∂₁∂^t₁, 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.