TY - JOUR
T1 - The non-commuting, non-generating graph of a nilpotent group
AU - Cameron, Peter J.
AU - Freedman, Saul Daniel
AU - Roney-Dougal, Colva Mary
N1 - Funding: 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).
PY - 2021/1/29
Y1 - 2021/1/29
N2 - 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.
AB - 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.
U2 - 10.37236/9802
DO - 10.37236/9802
M3 - Article
SN - 1077-8926
VL - 28
JO - Electronic Journal of Combinatorics
JF - Electronic Journal of Combinatorics
IS - 1
M1 - P1.16
ER -