Barycentric Characteristic Numbers

If G is the category of finite simple graphs G = (V,E), the linear space V of valuations on G has a basis given by the f-numbers vk(G) counting complete subgraphs Kk+1 in G. The barycentric refinement G1 of G ∈ G is the graph with Kl subgraphs as vertex set where new vertices a 6= b are connected if a ⊂ b or b ⊂ a. Under refinement, the clique data transform as ~v → A~v with the upper triangular matrix Aij = i!S(j, i) with Stirling numbers S(j, i). The eigenvectors χk of A T with eigenvalues k! form an other basis in V. The χk are normalized so that the first nonzero entry is > 0 and all entries are in Z with no common prime factor. χ1 is the Euler characteristic ∑∞ k=0(−1)vk, the homotopy and so cohomology invariant on G. Half of the χk will be zero Dehn-Sommerville-Klee invariants like half the Betti numbers are redundant under Poincaré duality. On the