Explicit Betti Numbers for Skeletons of Chordal Clique Complexes and Their Alexander Duals

Let $Δ_{\mathbf r}=Δ_{\mathbf r}(n_1,\ldots,n_e)$ be the clique complex of a chordal graph with maximal cliques $S_1,\ldots,S_e$ in a leaf order, where $n_m=|S_m|$, $r_m=|S_{m+1}\cap(S_1\cup\cdots\cup S_m)|$, $\mathbf r=(r_1,\ldots,r_{e-1})$, and $N_{\mathbf r}$ is the number of vertices. We determi