Graphs of relations and Hilbert series

Abstact We are discussing certain combinatorial and counting problems related to quadratic algebras. First we give examples which confirm the Anick conjecture on the minimal Hilbert series for algebras given by n generators and n(n−1) 2 relations for n 6 7. Then we investigate combinatorial structure of colored graph associated to relations of RIT algebra. Precise descriptions of graphs (maps) corresponding to algebras with maximal Hilbert series are given in certain cases. As a consequence it turns out, for example, that RIT algebra may have a maximal Hilbert series only if components of the graph associated to each color are pairwise 2-isomorphic.