On the inclusion ideal graph of semigroups

Abstract

The inclusion ideal graph In(S) of a semigroup S is an undirected simple graph whose vertices are all nontrivial left ideals of S and two distinct left ideals I,J are adjacent if and only if either I⊂J or J⊂I. The purpose of this paper is to study algebraic properties of the semigroup S as well as graph theoretic properties of In(S). In this paper, we investigate the connectedness of In(S). We show that diameter of In(S) is at most 3 if it is connected. We also obtain a necessary and sufficient condition of S such that the clique number of In(S) is n, where n is the number of minimal left ideals of S. Further, various graph invariants of In(S) viz. perfectness, planarity, girth etc. are discussed. For a completely simple semigroup S, we investigate various properties of In(S) including its independence number and matching number. Finally, we obtain the automorphism group of In(S).