On the Graphs Related to Green Relations of Finite Semigroups

In this paper we develop an analog of the notion of the con- jugacy graph of finite groups for the finite semigroups by considering the Green relations of a finite semigroup. More precisely, by defining the new graphs L(S), R(S), H(S), J (S) and D(S) (we name them the Green graphs) related to the Green relations L,R, J,H and D of a finite semigroup S, we first attempt to prove that the graphs L(S) and H(S) have exactly one connected component, and this graphs for regu- lar semigroups are complete. And secondly, we give a necessary condition for a finite semigroup to be regular. This study shows an intrinsic differ- ence between the conjugacy graphs (of groups) and the Green graphs (of semigroups) as well. Finally, our calculations include two kinds of semi- groups, mostly involving the well known Lucas numbers, and examining the proved assertions.