Line Graph Associated to Graph of a Near-Ring with Respect to an Ideal

Abstract

Let $N$ be a near-ring and $I$ be an ideal of $N$. The graph of $N$ with respect to $I$ is a graph with $V (N )$ as vertex set and any two distinct vertices $x$ and $y$ are adjacent if and only if $xNy \subseteq I$ or $yNx \subseteq I$. This graph is denoted by $G_I(N)$. We define the line graph of $G_I(N)$ as a graph with each edge of $G_I (N )$ as vertex and any two distinct vertices are adjacent if and only if their corresponding edges share a common vertex in the graph $G_I (N )$. We denote this graph by $L(G_I (N ))$. We have discussed the diameter, girth, clique number, dominating set of $L(G_I(N))$. We have also found conditions for the graph $L(G_I(N))$ to be acycle graph.