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

TL;DR

This paper introduces the coprime order graph of finite groups, exploring its properties and relations to group structure, including conditions for Eulerian, connected, complete, planar, and Hamiltonian graphs, with spectral analysis for specific groups.

Contribution

The paper defines a new coprime order graph for finite groups and investigates its properties, linking graph characteristics to group-theoretic features, and computes spectral data for particular cases.

Findings

Characterizes when the graph is Eulerian, connected, complete, planar, and Hamiltonian.

Provides spectral analysis of the signless Laplacian for specific groups.

Derives conditions for graph properties based on group parameters.

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 $m\in \mathbb{N}$.