Non-nilpotent graph of a group

الأصل · EN

We associate a graph NG 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 paper we study the graph theoretical properties of NG and its induced subgraph on G nil(G), where nil(G)={x∈ G | < x,y> is nilpotent for all y∈ G}. For any finite group G, we prove that NG has either |Z*(G)| or |Z*(G)|+1 connected components, where Z*(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 NG has at most two elements.