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 ∈ G | ⟨ x, y ⟩ is nilpotent for all y ∈ G}. For any finite group G, we prove that 𝒩 _G 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 𝒩 _G has at most two elements.

Key Words:

• Graphs associated to groups
• Hypercenter of a group
• Nilpotent group
• Regular graph

2000 Mathematics Subject Classification:

• 20D60
• 05C25

ACKNOWLEDGMENTS

The research of the first author was partially supported by the Center of Excellence for Mathematics, University of Isfahan, and he gratefully acknowledges the financial support of University of Isfahan for the sabbatical leave studies at University of Bath, UK, and ICTP, Trieste, Italy.

Notes

Communicated by A. Turull.