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 $$x,y\in G$$ 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].