This paper studies the zero-divisor graphs attached to several finite chain-ring families and computes the homological invariants of their edge ideals by using cochordal constructible systems. We begin with a general layered graph $C(q,L)$, whose vertices are arranged according to valuation layers and whose adjacency is governed by the single rule $k+\ell\ge L$, form some integers $k$ and $\ell$. This graph models the zero-divisor structure of a finite chain ring with residue field of order $q$ and nilpotency index $L$. We prove that $C(q,L)$ is cochordal, determine its type sequence, then correct and refine the Betti formula of its edge ideal [Dung and Vu, Cochordal zero divisor graphs and Betti numbers of their edge ideals, Comm. Algebra 54(2) (2026) 736--744]. The results are then specialized to the Gaussian quotient rings $\mathbb Z_{2^m}[i]$ and to the truncated polynomial rings $\mathbb Z_p[x]/(x^c)$. We compute projective dimension, regularity, independence number, height, Hilbert series, and Cohen--Macaulay behavior. The computations show that these quotient rings have $2$-linear resolutions, while Cohen--Macaulayness occurs only in the expected degenerate or complete-graph cases.

翻译：本文研究了与若干有限链环族相关的零因子图，并通过使用余弦图可构造系统计算其边理想的同调不变量。我们从一般分层图$C(q,L)$开始，其顶点根据赋值层排列，邻接关系由单一规则$k+\ell\ge L$（$k$和$\ell$为整数）决定。该图模拟了剩余域阶为$q$且幂零指数为$L$的有限链环的零因子结构。我们证明了$C(q,L)$是余弦图，确定了其类型序列，进而修正并完善了其边理想的贝蒂公式[Dung and Vu, Cochordal zero divisor graphs and Betti numbers of their edge ideals, Comm. Algebra 54(2) (2026) 736--744]。随后将结果特化到高斯商环$\mathbb Z_{2^m}[i]$和截断多项式环$\mathbb Z_p[x]/(x^c)$。我们计算了射影维数、正则度、独立数、高度、希尔伯特级数及科恩-马考莱性质。计算表明，这些商环具有$2$-线性分解，而科恩-马考莱性仅在预期的退化情形或完全图情形中出现。