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

Abstract

In this paper, a new graph structure called the \textit{coprime order graph} of a finite group $G$ denoted by $\Theta(G)$ has been introduced. The \textit{coprime graph} of a finite group introduced by Ma, Wei, and Yang [\textit{The coprime graph of a group. International Journal of Group Theory, 3(3), pp.13-23.}] is a subgraph of the \textit{coprime order graph} introduced in this paper. The vertex set of $\Theta(G)$ is $G$, and any two vertices $x,y$ in $\Theta(G)$ are adjacent if and only if $\gcd(o(x),o(y))$ is equal to $1$ or a prime number. We study how the graph properties of $\Theta(G)$ and group properties of $G$ are related among themselves. We provide a necessary and sufficient condition for $\Theta(G)$ to be Eulerian for any finite group $G$. We also study $\Theta(G)$ for certain finite groups like $\mathbb Z_n$ and $\mbox D_n$ and derive conditions when it is connected, complete, planar, and Hamiltonian for various $n\in \mathbb N$. We also study the vertex connectivity of $\Theta(\mathbb Z_n)$ for various $n\in \mathbb N.$ Finally, we have computed the signless Laplacian spectrum of $\Theta(G)$ when $G=\mathbb Z_n$ and $G=\mbox D_n$ for $n\in \{pq,p^m\}$ where $p,q$ are distinct primes and $m\in \mathbb{N}$.