The non-commuting, non-generating graph of a nilpotent group
MetadataShow full item record
Altmetrics Handle Statistics
Altmetrics DOI Statistics
For a nilpotent group G, let nc(G) be the difference between the complement of the generating graph of G and the commuting graph of G, with vertices corresponding to central elements of G removed. That is, nc(G) has vertex set G \ Z(G), with two vertices adjacent if and only if they do not commute and do not generate G. Additionally, let nd(G) be the subgraph of nc(G) induced by its non-isolated vertices. We show that if nc(G) has an edge, then nd(G) is connected with diameter 2 or 3, with nc(G) = nd(G) in the diameter 3 case. In the infinite case, our results apply more generally, to any group with every maximal subgroup normal. When G is finite, we explore the relationship between the structures of G and nc(G) in more detail.
Cameron , P J , Freedman , S D & Roney-Dougal , C M 2021 , ' The non-commuting, non-generating graph of a nilpotent group ' , Electronic Journal of Combinatorics , vol. 28 , no. 1 , P1.16 . https://doi.org/10.37236/9802
Electronic Journal of Combinatorics
Copyright © The authors. Released under the CC BY-ND license (International 4.0).
DescriptionFunding: UK ESPRC grant number EP/R014604/1, and partially supported by a grant from the Simons Foundation (PJC, CMR-D); ESPRC grant number EP/R014604/1, St Leonard’s International Doctoral Fees Scholarship, School of Mathematics & Statistics PhD Funding Scholarship at the University of St Andrews (SDF).
Items in the St Andrews Research Repository are protected by copyright, with all rights reserved, unless otherwise indicated.