The graded annihilating submodule graph

In this paper, we study the graded annihilating graph for submodules, representing graded submodules as vertices connected by edges following a specific pattern. Our exploration leads us through the intricacies of this graph, uncovering insights into their connectivity, girth, bipartition, and completeness within graded modules, establishing connections between these graded annihilating graph submodules and their ungraded counterparts. We also extend prior work on finite girth conditions of annihilating graphs for modules. Our contributions include two comprehensive criteria for graded modules (Theorem 3.7 and Corollary 3.8), which also apply to non-graded modules with trivial gradation. These findings have implications for this article and future research.