A parametrized Poincare-Hopf Theorem and Clique Cardinalities of graphs

Given a locally injective real function g on the vertex set V of a finite simple graph G=(V,E), we prove the Poincare-Hopf formula fG(t) = 1+t sum{x in V} f{Sg(x)}(t), where Sg(x) = { y in S(x), g(y) less than g(x) } and fG(t)=1+f0 t + ... + f{d} t^{d+1} is the f-function encoding the f-vector of a graph G, where fk counts the number of k-dimensional cliques, complete sub-graphs, in G. The corresponding computation of f reduces the problem recursively to n tasks of graphs of half the size. For t=-1, the parametric Poincare-Hopf formula reduces to the classical Poincare-Hopf result X(G)=sumx ig(x), with integer indices ig(x)=1-X(Sg(x)) and Euler characteristic X. In the new Poincare-Hopf formula, the indices are integer polynomials and the curvatures Kx(t) expressed as index expectations Kx(t) = E[ix(t)] are polynomials with rational coefficients. Integrating the Poincare-Hopf formula over probability spaces of functions g gives Gauss-Bonnet formulas like fG(t) = 1+sum{x} F{S(x)}(t), where FG is the anti-derivative of fG. A similar computation is done for the generating function f{G,H}(t,s) = sum{k,l} f{k,l}(G,H) s^k t^l of the f-intersection matrix f{k,l}(G,H) counting the number of intersections of k-simplices in G with l-simplices in H. Also here, the computation is reduced to 4 n² computations for graphs of half the size: f{G,H}(t,s) = sum{v,w} f{Bg(v),Bg(w)}(t,s) - f{Bg(v),Sg(w)}(t,s) - f{Sg(v),Bg(w)}(t,s) + f{Sg(v),Sg(w)}(t,s), where Bg(v)= S_g(v)+{v} is the unit ball of v.