NOTES ON THE CO-PRIME ORDER GRAPH OF A GROUP

The co-prime order graph of a group G is the graph with vertex set G, and two distinct elements x, y is an element of G are adjacent if gcd(o(x), o(y)) is either 1 or a prime, where o(x) and o(y) are the orders of x and y, respectively. In this paper, we characterize finite groups whose co-prime order graphs are complete and classify finite groups whose co-prime order graphs are planar, which generalizes some results by BANERJEE [(3)]. We also compute the vertex-connectivity of the co-prime order graph of a cyclic group, a dihedral group and a generalized quaternion group, which answers a question by Banerjee [(3)].