Betti Numbers of Sequentially Cohen-Macaulay Co-Chordal Graphs and Their Applications

Abstract

We study sequentially Cohen-Macaulay co-chordal graphs through the glued clique complexes of their chordal complements. Using the characterization of these complements as $(d_1,\ldots,d_q)$-trees, we derive explicit formulas for the graded Betti numbers of the associated edge ideals, yielding a complete homological characterization of sequentially Cohen-Macaulay co-chordal graphs. As applications, we determine exact homological invariants for several important graph families, including split graphs, threshold graphs, and prime ideal graphs, and classify their Cohen-Macaulay cases. We further characterize the sequentially Cohen-Macaulay nilpotent graphs of finite direct products of Artinian chain rings and establish a closed formula for their graded Betti numbers in terms of local nilpotency indices and residue field cardinalities. Finally, we classify the zero-divisor graphs of $\mathbb{Z}_n$, proving that they are sequentially Cohen-Macaulay if and only if $n=2p$ or $n=p^a$, where $p$ is a prime.