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

Abstract

Let $\Delta_{\mathbf r}=\Delta_{\mathbf r}(n_1,\ldots,n_e)$ 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 determine the graded Betti numbers $\beta_{i,j}\bigl(\mathbb K[(\Delta_{\mathbf r})_{(k)}]\bigr)$ of every skeleton and derive explicit formulas for its regularity, projective dimension, depth, multiplicity, Cohen-Macaulayness, and initially Cohen-Macaulayness. We also compute its extremal Betti numbers and describe its integral homology and homotopy type. For the Alexander dual $\Delta_{\mathbf r}^{\vee}$, we construct an explicit minimal multigraded resolution whose shifts are determined by $N_{\mathbf r}-n_m$ and $N_{\mathbf r}-r_m$, and obtain its graded and multigraded Betti numbers, canonical module, Cohen-Macaulay type, $a$-invariant, and Gorenstein criterion. We further show that every proper skeleton $(\Delta_{\mathbf r}^{\vee})_{(k)}$ is Cohen-Macaulay and level, determine its Betti numbers and type, and characterize its Gorenstein cases. Finally, we show that the graded Betti table of $\mathbb K[\Delta_{\mathbf r}^{\vee}]$ determines the multisets ${n_m}$ and ${r_m}$ up to a common shift.