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Betti numbers of Stanley--Reisner rings with pure resolutions

Let $Δ$ be simplicial complex and let $k[Δ]$ denote the Stanley--Reisner ring corresponding to $Δ$. Suppose that $k[Δ]$ has a pure free resolution. Then we describe the Betti numbers and the Hilbert--Samuel multiplicity of $k[Δ]$ in terms of the $h$--vector of $Δ$. As an application, we derive a linear equation system for the components of the $h$--vector of the clique complex of an arbitrary chordal graph.