THE CODIVISOR GRAPH OF A FINITE RING WITH UNITY

Abstract

For a finite ring with unity $R$, we define a simple undirected graph called the codivisor graph $\Psi(R)$ with all the non-zero elements of the ring $R$ as vertices and two distinct vertices $a$ and $b$ are adjacent to each other if and only if $a \mid b$ and $b \mid a$. We first consider its connectedness. Looking at $\mathbb{Z}_n$, we determine the condition for connectedness of $\Psi\left(\mathbb{Z}_n\right)$ and also discuss its structure. We further investigate properties like diameter, girth, completeness, planarity, traversability, independence number and domination number in our work.

Received: March 4, 2023;
Revised: April 28, 2023;
Accepted: May 12, 2023;