Non-Nilpotent Graph of a Group

Abstract

We associate a graph N G with a group G (called the non-nilpotent graph of G) as follows: take G as the vertex set and two vertices are adjacent if they generate a non-nilpotent subgroup. In this article, we study the graph theoretical properties of N G and its induced subgraph on G \ nil(G), where nil(G) = {x ε G {pipe}(x, y)is nilpotent for all y ε G}. For any finite group G, we prove that N G has either {pipe} Z.ast;(G){pipe} or {pipe}Z.ast;(G){pipe} +1 connected components, where Z.ast;(G) is the hypercenter of G. We give a new characterization for finite nilpotent groups in terms of the non-nilpotent graph. In fact, we prove that a finite group G is nilpotent if and only if the set of vertex degrees of N G has at most two elements. © 2010 Copyright Taylor and Francis Group, LLC.