Intersection non-simple graph of an abelian group

The intersection non-simple graph, denoted by INS(G), of a finite abelian group G is an undirected graph whose vertex set is the collection of all proper non-trivial subgroups of G, and any two distinct vertices are adjacent if and only if their intersection is not a simple subgroup of G. We obtain some properties of INS(G) related to connectedness, completeness, degree, and girth. The concepts of bipartiteness, triangle-free, cluster, claw-free and cograph are taken into consideration. We also investigate the clique number, independence number, domination number, and planarity of INS(G).