The $f$--vector of the clique complex of chordal graphs and Betti numbers of edge ideals of uniform hypergraphs

We describe the Betti numbers of the edge ideals $I(G)$ of uniform hypergraphs $G$ such that $I(G)$ has linear graded free resolution. We give an algebraic equation system and some inequalities for the components of the $f$–vector of the clique complex of an arbitrary chordal graph. Finally we present an explicit formula for the multiplicity of the Stanley-Reisner ring of the edge ideals of any chordal graph.