Non-exterior square graph of finite group

Abstract

We define the non-exterior square graph ??G which is a graph associated to a non-cyclic finite group with the vertex set G\?Z(G), where ?Z(G) denotes the exterior centre of G, and two vertices x and y are joined whenever x ^ y ? 1, where ^ denotes the operator of non-abelian exterior square. In this paper, we investigate how the group structure can be affected by the planarity, completeness and regularity of this graph.