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 khk of A T with eigenvalues k! form an other basis in V. The khk are normalized so that the first nonzero entry is 0 and all entries are in Z with no common prime factor. kh1 is the Euler characteristic k=0( 1)vk, the homotopy and so cohomology invariant on G. Half of the khk will be zero Dehn Sommerville Klee invariants like half the Betti numbers are redundant under Poincare duality. On the