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

We study the homological properties of ${\mathrm{\Delta }}_{\mathbf{r}}(n_1,\dots ,n_e)$, a simplicial complex formed by sequentially gluing complete graphs along $(r_i-1)$-simplices. This construction generates precisely the chordal clique complexes, whose Stanley-Reisner ideals admit 2-linear resolutions. By computing the $f$-vector and evaluating the Hilbert series, we establish explicit graded Betti numbers for all $k$-skeletons. We show that the regularity of these skeletons is $k+1$ and the projective dimension stabilizes at $N_{\mathbf{r}}-r_{\min }-1$ for $k\ge r_{\min }$, providing a complete classification of when the complex is Cohen-Macaulay, sequentially Cohen-Macaulay, or initially Cohen-Macaulay. We also obtain explicit formulas for the ring multiplicity and reduced Euler characteristic. Applying Alexander duality, we derive the $f$-vector, rational $h$-polynomial, and exact graded Betti numbers of the dual and its skeletons. Furthermore, analyzing these dual skeletons yields a family of complexes that resolve recent open bounds on regularity. Finally, equating the topological and rational evaluations of the Hilbert series produces a new family of combinatorial binomial identities.