Curvature from Graph Colorings

Given a finite simple graph G=(V,E) with chromatic number c and chromatic polynomial C(x). Every vertex graph coloring f of G defines an index i_f(x) satisfying the Poincare-Hopf theorem sum_x i_f(x)=chi(G). As a variant to the index expectation result we prove that E[i_f(x)] is equal to curvature K(x) satisfying Gauss-Bonnet sum_x K(x) = \chi(G), where the expectation is the average over the finite probability space containing the C(c) possible colorings with c colors, for which each coloring has the same probability.