Gauss-Bonnet for multi-linear valuations

Abstract

We prove Gauss-Bonnet and Poincare-Hopf formulas for multi-linear valuations on finite simple graphs G=(V,E) and answer affirmatively a conjecture of Gruenbaum from 1970 by constructing higher order Dehn-Sommerville valuations which vanish for all d-graphs without boundary. An example of a quadratic valuation is the Wu characteristic w(G) which sums (-1)^(dim(x)+dim(y)) over all intersecting pairs of complete subgraphs x,y of a G. More generally, an intersection number w(A,B) sums (-1)^(dim(x)+dim(y)) over pairs x,y, where x is in A and y is in B and x,y intersect. w(G) is a quadratic Euler characteristic X(G), where X sums (-1)^dim(x) over all complete subgraphs x of G. We prove that w is multiplicative, like Euler characteristic: w(G x H) = w(G) w(H) for any two graphs and that w is invariant under Barycentric refinements. We construct a curvature K satisfying Gauss-Bonnet w(G) = sum K(a). We also prove w(G) = X(G)-X(dG) for Euler characteristic X which holds for any d-graph G with boundary dG. We also show higher order Poincare-Hopf formulas: there is for every multi-linear valuation X and function f an index i(a) such that sum i(a)=X(G). For d-graphs G and X=w it agrees with the Euler curvature. For the vanishing multi-valuations which were conjectured to exist, like for the quadratic valuation X(G) = (V X) Y with X=(1,-1,1,-1,1),Y=(0,-2,3,-4,5) on 4-graphs, discrete 4 manifolds, where V_{ij}(G) is the f-matrix counting the number of i and j-simplices in G intersecting, the curvature is constant zero. For all graphs and multi-linear Dehn-Sommerville relations, the Dehn-Sommerville curvature K(v) at a vertex is a Dehn-Sommerville valuation on the unit sphere S(v). We show X V(G) Y = v(G) Y for any linear valuation Y of a d-graph G with f-vector v(G). This provides examples for the Gruenbaum conjecture.