Intersection Operator Graph of a Group

Abstract

Let $(G,*)$ be a group with binary operation $\lq*\rq$. The Intersection Operator graph $\ig(G)$ of $G$ is a graph with $V(\ig(G)) = G$ and two distinct vertices $x$ and $y$ are adjacent in $\ig(G)$ if and only if $\langle x\rangle \cap \langle y\rangle \subseteq \langle x*y\rangle $. In this paper, we want to explore how the group theoretical properties of $G$ can effect on the graph theoretical properties of $\ig(G)$. Some characterizations for fundamental properties of $\ig(G)$ have also been obtained. Finally, we characterize certain classes of Intersection Operator Graph corresponding to finite abelian groups.