Proper vertex colorings of a graph are related to its boundary map
∂
1
\partial _1
, also called its signed vertex-edge incidence matrix. The vertex Laplacian of a graph,
L
=
∂
1
∂
1
t
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.