Reduced power graph of a group

Let S be a semigroup. We define the directed reduced power graph of S, denoted by P→(S), is a digraph with vertex set S, and for u, v ∈ S, there is an arc from u to v if and only if u ≠ v and 〈v〉⊂〈u〉. The (undirected) reduced power graph of S, denoted by P(S), is the underlying graph of P→(S). This means that the set of vertices of P(S) is equal to S and two vertices u and v are adjacent if and only if u ≠ v and 〈v〉⊂〈u〉 or 〈u〉⊂〈v〉. In this paper, we study some interplay between the algebraic properties of a group and the graph theoretic properties of its (directed and undirected) reduced power graphs. Also we establish some relationship between the reduced power graphs and power graphs of groups.