Non-nilpotent Graph of a Group
نویسنده
چکیده
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.
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