On the intersection ideal graph of semigroups

Abstract

The intersection ideal graph Γ(S) of a semigroup S is a simple undirected graph whose vertices are all nontrivial left ideals of S and two distinct left ideals I, J are adjacent if and only if their intersection is nontrivial.In this paper, we investigate the connectedness of Γ(S). We show that if Γ(S) is connected then diam(Γ(S)) ≤ 2.Further we classify the semigroups such that the diameter of their intersection graph is two. Other graph invariants, namely perfectness, planarity, girth, dominance number, clique number, independence number etc. are also discussed.Finally, if S is union of n minimal left ideals then we obtain the automorphism group of Γ(S).Recently, Chakrabarty et al. [8] introduced the notion of intersection ideal graph of rings. The intersection ideal graph Γ(R) of a ring R is an undirected simple graph whose vertex set is the collection of nontrivial left ideals of R and two distinct vertices I, J are adjacent if and only if I ∩ J = {0}. They characterized the rings R for which the graph Γ(R) is connected and obtain several necessary and sufficient conditions on a ring R such that Γ(R) is complete. Planarity of intersection graphs of ideals of ring with unity is described in [15] and domination number in 2010 Mathematics Subject Classification. 05C25.