Generalized binomial edge ideals of complete $r$-partite graphs

This paper analyzes the cohomological dimension of the generalized binomial edge ideal ${\text{calJ}}_{K_m,G}$ for a complete $r$-partite graph $G$. Additionally, the Krull dimension, the depth, the Castelnuovo–Mumford regularity, the Hilbert series, and the multiplicity of its quotient ring are explicitly determined.