Abstract: Let $$Q$$ be a group with identity $$e$$, and let $$R = \bigoplus_{q \in Q} R_q$$ be a $$Q$$-graded ring with unity. We introduce the graded-simple-intersection graph $$GS_Q(R)$$, whose vertices are the nonzero graded ideals of $$R$$, where two distinct vertices are adjacent precisely when their intersection is a nonzero graded simple ideal. This graph naturally is a subgraph of the graded-ideal intersection graph by requiring intersections to be graded simple rather than merely nonzero. We investigate how the algebraic structure of a graded ring $$R$$ influences the graph-theoretic properties of $$GS_Q(R)$$. In particular, we characterize the connectedness of $$GS_Q(R)$$, determine its possible girth and diameter, prove that its diameter is never equal to 3, and establish necessary and sufficient conditions for $$GS_Q(R)$$ to be bipartite. We also characterize when $$GS_Q(R)$$ is complete and when it is complete bipartite, revealing further connections between the graded ring structure and the associated graph.

Keywords: graded simple-intersection graph, graded ring, graded ideal, girth, diameter, bipartite graph

Pages: 21-30