Some results on the reduced power graph of a group

The reduced power graph RP(G) of a group G is the graph with vertex set G and two vertices u and v are adjacent if and only if v ⊂ u or u ⊂ v. The proper reduced power graph RP^*(G) of G is the subgraph of RP(G) induced on G \. In this paper, we classify the finite groups whose reduced power graph (resp. proper reduced power graph) is one of complete k-partite, acyclic, triangle free or claw-free (resp. complete k-partite, acyclic, triangle free, claw-free, star or tree). In addition, we obtain the clique number and the chromatic number of RP(G) and RP^*(G) for any torsion group G. Also, for a finite group G, we determine the girth of RP^*(G). Further, we discuss the cut vertices, cut edges and perfectness of these graphs. Then we investigate the connectivity, the independence number and the Hamiltonicity of the reduced power graph (resp. proper reduced power graph) of some class of groups. Finally, we determine the number of components and the diameter of RP^*(G) for any finite group G.