The order divisor-power graph of finite groups

Let G be a finite group. In this paper, we introduce the order divisor-power graph Γodp(G) associated with G as the simple undirected graph whose vertices are the elements of G and such that two vertices a, b a̸ = b are adjacent if one is a power of the other and their orders are different. We investigate some algebraic properties and combinatorial structures of the order divisor-power graph Γodp(G) and obtain the conditions under which the order divisor-power graph Γodp(G) can be a star graph. Also, we exhibit some connection between the order divisor-power graph and the power graph of dihedral groups up to an isomorphism. Furthermore, we prove that the order divisor-power graphs of some classes of dihedral groups are neither bipartite nor tripartite, but it is a complete multipartite graph if the group is a cyclic group.