This article is mainly concerned with computations of some algebraic invariants of quotient rings of edge ideals of perfect $[h,d]$-ary trees and unicyclic graphs. We compute exact values of depth and Stanley depth and consequently projective dimension for above mentioned quotient rings, except for the one special case of unicyclic graph for which best possible bounds of Stanley depth are given.

This paper investigates algebraic invariants of edge ideals associated with families of graphs constructed as the strong product of a path or cycle with the complete graph $K_m$, namely ${\mathcal{P}}_{\gamma }=P_{\gamma }⊠K_m$ and ${\mathcal{C}}_{\gamma }=C_{\gamma }⊠K_m$. For an edge ideal $I(\mathbb{G})\subset \mathrm{\Re }$ in a polynomial ring over a field $\mathbb{K}$, we derive explicit combinatorial formulas for key homological and ring-theoretic invariants of the quotient ring $\mathrm{\Re }/I(\mathbb{G})$. These include Castelnuovo-Mumford regularity, depth, Stanley depth, projective dimension, and Krull dimension. Furthermore, we characterize all Cohen–Macaulay graphs in defined families, providing a complete classification where $\mathrm{\Re }/I(\mathbb{G})$ is Cohen–Macaulay.