Betti numbers of sequentially cohen macaulay co-chordal graphs and their applications

We study Betti numbers of sequentially Cohen–Macaulay co-chordal graphs through the maximal cliques of their chordal complements. Starting from the known [Formula: see text]-tree characterization and the Hilbert series formula for chordal clique complexes, we prove a converse Betti criterion showing that a co-chordal graph is sequentially Cohen–Macaulay precisely when a nonincreasing clique gluing order satisfies a specific binomial identity. We also characterize the Cohen–Macaulay case by the Betti sequence [Formula: see text], where [Formula: see text] is the number of maximal cliques of the complement. Applications are given to split graphs, prime ideal graphs, and threshold graphs. For finite products of Artinian chain rings, we prove that the nilradical graph is threshold, or equivalently co-chordal with a sequentially Cohen–Macaulay edge ring, exactly when at most one factor has nilpotency index at least three. We also give its Betti numbers explicitly. Finally, within the known co-chordal range for zero divisor graphs of [Formula: see text], we prove that the sequentially Cohen–Macaulay cases are precisely [Formula: see text], [Formula: see text], and [Formula: see text], with [Formula: see text] odd in the last two families.