Structural Properties of the Simple-Intersection Graph of Graded Rings

Let Q be a group with identity e , and let R = ⨁ q ∈ Q R q be a Q -graded ring with unity. We introduce the graded-simple-intersection graph G S 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 G S Q ( R ) . In particular, we characterize the connectedness of G S Q ( R ) , determine its possible girth and diameter, prove that its diameter is never equal to 3, and establish necessary and sufficient conditions for G S Q ( R ) to be bipartite. We also characterize when G S Q ( R ) is complete and when it is complete bipartite, revealing further connections between the graded ring structure and the associated graph.