Non-Exterior Square Graph of Finite Group

Abstract

We define the non-exterior square graph $\hat{\Gamma}_{G}$
which is a graph associated to a non-cyclic group with the vertex set $G\setminus \hat{Z}(G)$, and two vertices $x$ and $y$ join whenever $x\wedge y\neq 1$. In this paper, we investigate how the group structure can affect on the planarity, completeness and regularity of this graph.