On a New Graph defined on the order of elements of a Finite Group

Subarsha Banerjee

Published 2019-11-07Version 1

In this paper, the author has introduced a new graph structure called the \textbf{Co-Prime Order Graph} $\Theta(G)$ on a finite group $G$ whose vertex set is $G$ and any two vertexes $x,y$ in $\Theta(G)$ are adjacent if and only if $\gcd(o(x),o(y))=1$ or prime. We study how the graph properties of $\Theta(G)$ and group properties of $G$ are related among themselves. We have given various conditions when $\Theta(G)$ is connected, complete, planar and Hamiltonian when $G=\Bbb Z_n$ and $G=D_n$ for various $n\in \Bbb N$. We have also found when the graph $\Theta(G)$ is Eulerian for any finite group $G$. We have also studied the vertex connectivity of $\Theta(\Bbb Z_n)$ for various $n\in \Bbb N.$ Finally we have computed the Signless Laplacian Spectra of $\Theta(G)$ when $G=\Bbb Z_n$ and $G=D_n$ for $n=pq$ and $n=p^m$ where $p,q$ are primes and $m\in \Bbb Z$.