Algebraic invariants of edge ideals from strong products of paths and cycles

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.