Curvature from Graph Colorings

Given a finite simple graphG = (V,E) with chromatic number c and chromatic polynomial C(x). Every vertex graph coloring f of G defines an index if (x) satisfying the Poincaré-Hopf theorem [17] ∑ x if (x) = χ(G). As a variant to the index expectation result [19] we prove that E[if (x)] is equal to curvature K(x) satisfying Gauss-Bonnet ∑ xK(x) = χ(G) [16], 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.