Line graph associated to the intersection graph of ideals of rings

Abstract

Let R be a ring with unity and I(R)^∗ are all non-trivial left ideals of R. The intersection graph of ideals of R is denoted by G(R) is an undirected simple graph with vertex set I(R)^∗ and two distinct vertices I and J are adjacent if and only if I∩J ≠ 0. In this article, we investigate some basic properties of the line graph associated to G(R), denoted by L(G(R)). Moreover, we investigate completeness, unicyclicness, bipartiteness, planarity, outerplanarity, ring graph, diameter, girth and clique of L(G(Z_n)). We also investigate some basic properties of L(G(R)) for left Artinian ring and finally, we determine the domination number and bondage number of L(G(Z_n)).