TY - JOUR
T1 - The non-commuting, non-generating graph of a nilpotent group
AU - Cameron, Peter J.
AU - Freedman, Saul D.
AU - Roney-Dougal, Colva M.
N1 - Funding Information:
∗Supported by ESPRC grant number EP/R014604/1, and partially supported by a grant from the Simons Foundation.
Funding Information:
†Supported by ESPRC grant number EP/R014604/1, by a St Leonard’s International Doctoral Fees Scholarship, and by a School of Mathematics & Statistics PhD Funding Scholarship at the University of St Andrews.
Publisher Copyright:
© The authors.
PY - 2021/1/29
Y1 - 2021/1/29
N2 - For a nilpotent group G, let Ξ(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, Ξ(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 Ξ+(G) be the subgraph of Ξ(G) induced by its non-isolated vertices. We show that if Ξ(G) has an edge, then Ξ+(G) is connected with diameter 2 or 3, with Ξ(G) = Ξ+(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 Ξ(G) in more detail.
AB - For a nilpotent group G, let Ξ(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, Ξ(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 Ξ+(G) be the subgraph of Ξ(G) induced by its non-isolated vertices. We show that if Ξ(G) has an edge, then Ξ+(G) is connected with diameter 2 or 3, with Ξ(G) = Ξ+(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 Ξ(G) in more detail.
UR - https://www.scopus.com/pages/publications/85100181093
U2 - 10.37236/9802
DO - 10.37236/9802
M3 - Article
AN - SCOPUS:85100181093
SN - 1077-8926
VL - 28
JO - The Electronic Journal of Combinatorics
JF - The Electronic Journal of Combinatorics
IS - 1
M1 - P1.16
ER -