The co-prime order graph associated with a finite group

Abstract

Let $G$ be a finite group. The co-prime order graph of $G$ is the graph whose vertex set is $G$, and two distinct vertices $x,y$ 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 all finite groups whose co-prime order graphs are complete and classify all finite groups whose co-prime order graphs are planar. Also, we 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 (2019). Finally, we prove that, for a fixed positive integer $k$, there are finitely many finite groups whose co-prime order graphs have (non)orientable genus $k$. As applications, we classify all finite groups whose co-prime order graphs have (non)orientable genus one and two.

arXiv commentThere is an error in Sect. 4. We can not obtain the first main result of this section

References in corpus (1)

• On a new graph defined on the order of elements of a finite group

Cited by in corpus (1)

• Enhanced Power Graph of Certain Non-abelian Groups