A few remarks on the theory of non-nilpotent graphs

Abstract

Abstract–We prove a few results about non-nilpotent graphs of symmetric groups S_n – namely that they satisfy a conjecture of Nongsiang and Saikia (which is likewise proved for alternating groups A_n), and that for $n⩾19$ each vertex has degree at least $\frac{n!}{2}$. We also show that the class of non-nilpotent graphs does not have any “local” properties, i.e. for every simple graph X there is a group G, such that its non-nilpotent graph contains X as an induced subgraph.

Keywords:

• Graphs associated to groups
• induced subgraphs
• nilpotent group
• symmetric group

2020 Mathematics Subject Classification:

• 20D60
• 05C25

Disclosure statement

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.