Generalized order divisor graphs of finite group

Let G be a finite group and k a fixed positive integer. We define the generalized order divisor graph of G to be a graph whose vertex set is the group G and in which two vertices a and b are adjacent if and only if the orders o(a^k) and o(b^k) are different and either o(a^k) divides o(b^k) or o(b^k) divides o(a^k). This generalizes the order divisor graphs of finite groups. Some properties of our graph are introduced, and we investigate the structure of the generalized order divisor graphs of finite cyclic groups.

Keywords:

Power graph , star graph , cyclic group