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Abstract: Proper vertex colorings of a graph are related to its boundary map, alsocalled its signed vertex-edge incidence matrix. The vertex Laplacian of agraph, a natural extension of the boundary map, leads us to introducenowhere-harmonic colorings and analogues of the chromatic polynomial andStanley-s theorem relating negative evaluations of the chromatic polynomial toacyclic orientations. Further, we discuss some examples demonstrating thatnowhere-harmonic colorings are more complicated from an enumerative perspectivethan proper colorings.

Autor: Matthias Beck, Benjamin Braun


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