INTERSECTION GRAPH OF $\gamma$-SETS IN THE TOTAL GRAPH OF $\mathbb{Z}_n$ WITH RESPECT TO NIL IDEAL

Abstract

For any non-reduced ring $\mathbb{Z}_n$, the total graph of $\mathbb{Z}_n$ with respect to nil ideal, denoted by $T\left(\Gamma_N\left(\mathbb{Z}_n\right)\right)$, is a simple, undirected graph having vertex set $\mathbb{Z}_n$ and any two distinct vertices $x$ and $y$ of $T\left(\Gamma_N\left(\mathbb{Z}_n\right)\right)$ are adjacent if and only if $x+y \in N\left(\mathbb{Z}_n\right)$, where $N\left(\mathbb{Z}_n\right)=\left\{x \in \mathbb{Z}_n: x^2=0\right\}$ denotes the nil ideal of $\mathbb{Z}_n$. In this paper, we introduce a new class of graphs called intersection graphs. The intersection graph of $\gamma$-sets of $T\left(\Gamma_N\left(\mathbb{Z}_n\right)\right)$, denoted by $I_N\left(\mathbb{Z}_n\right)$, is a simple, undirected graph in which all the $\gamma$-sets of $T\left(\Gamma_N\left(\mathbb{Z}_n\right)\right)$ are taken as vertices and any two distinct vertices $S_1$ and $S_2$ are adjacent if and only if they have non-empty intersection, i.e., $S_1 \cap S_2 \neq \phi$. We show that the intersection graph is Eulerian for every non-reduced $\mathbb{Z}_n$. We also characterize the values of $n$ for which the graph is planar. We also obtain the diameter, girth, domination, independence and clique numbers of these graphs.

Received: November 15, 2021
Accepted: January 24, 2022