For any non-reduced ring  the total graph of  with respect to nil ideal, denoted by is a simple, undirected graph having vertex set and any two distinct vertices x and y of are adjacent if and only if where denotes the nil ideal of  In this paper, we introduce a new class of graphs called intersection graphs. The intersection graph of g-sets of  denoted by  is a simple, undirected graph in which all the g-sets of  are taken as vertices and any two distinct vertices  and  are adjacent if and only if they have non-empty intersection, i.e.,  We show that the intersection graph is Eulerian for every non-reduced  We also characterize the values of n for which the graph is planar. We also obtain the diameter, girth, domination, independence and clique numbers of these graphs.

total graph, nil ideal, domination number, independence number, intersection graph.