Cubic graphical regular representations of PSU3(q)

Li, Jing Jian; Xia, Binzhou; Zhang, Xiao Qian; Zheng, Shasha

doi:10.1016/j.disc.2022.112982

Cited by 3 publications

(4 citation statements)

References 13 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…S. Zheng [2] estimate the number of GRRs of a given group with large enough order and confirm the conjecture of Babai, Godsil, Imrich and Lovász as well as the conjecture of Xu. Some of the research has been published in [2,6] or can be found in [5].…”

supporting

confidence: 77%

“…In 1982, Babai, Godsil, Imrich and Lovász conjectured that except for Cayley graphs of two infinite families of groups—abelian groups of exponent greater than and generalised dicyclic groups—almost all finite Cayley graphs are GRRs (see [1, Conjecture 2.1]). In the study of GRRs of valency three, Xia and Fang [4] conjectured that with finitely many exceptions, every finite nonabelian simple group has a cubic GRR; and Spiga [3] conjectured that except for two-dimensional projective special linear groups and a finite number of other cases, every finite nonabelian simple group admits a cubic GRR with connection set containing only one involution. In 1998, Xu [7] introduced the concept of normal Cayley graphs: a Cayley graph of a group is called normal if the right regular permutation representation of the group is normal in the full automorphism group of the graph.…”

mentioning

confidence: 99%

See 1 more Smart Citation

Cayley Graphs and Graphical Regular Representations

ZHENG

2023

Bull. Aust. Math. Soc.

Self Cite

View full text Add to dashboard Cite

Studying highly symmetric graphs, especially vertex-transitive graphs, is an interesting and challenging topic in algebraic graph theory. In this thesis, we focus our attention on Cayley graphs, a family of vertex-transitive graphs. A graphical regular representation (GRR for short) of a group G is a Cayley graph of G whose full automorphism group is equal to the right regular permutation representation of G.In 1982, Babai, Godsil, Imrich and Lovász conjectured that except for Cayley graphs of two infinite families of groups-abelian groups of exponent greater than 2 and generalised dicyclic groups-almost all finite Cayley graphs are GRRs (see [1, Conjecture 2.1]). In the study of GRRs of valency three, Xia and Fang [4] conjectured that with finitely many exceptions, every finite nonabelian simple group has a cubic GRR; and Spiga [3] conjectured that except for two-dimensional projective special linear groups and a finite number of other cases, every finite nonabelian simple group admits a cubic GRR with connection set containing only one involution. In 1998, Xu [7] introduced the concept of normal Cayley graphs: a Cayley graph of a group is called normal if the right regular permutation representation of the group is normal in the full automorphism group of the graph. In [7], Xu conjectured that except for Cayley graphs of Hamiltonian 2-groups, almost all finite Cayley graphs are normal. The purpose of this thesis is to study these problems.The main work of this thesis consists of two parts, which are presented in Chapter 3 and Chapter 4, respectively. In the first part, we study cubic GRRs of some families of classical groups and, based on some previously known results, confirm the conjecture of Fang and Xia, and a modified version of Spiga's conjecture. In the second part, we

show abstract

supporting

confidence: 77%

mentioning

confidence: 99%

Cayley Graphs and Graphical Regular Representations

ZHENG

2023

Bull. Aust. Math. Soc.

Self Cite

View full text Add to dashboard Cite

show abstract

“…As mentioned above, the finite nonabelian simple groups form a large class of (2, p)generated groups, and the special interest in the existence of GRRs of nonabelian simple groups has been in the cubic case. For example, it was conjectured in [21] and recently proved in [22] that, except for a finite number of cases, every finite nonabelian simple group has a cubic GRR. Then a natural conjecture to extend this is: Conjecture 1.4.…”

Section: Introductionmentioning

confidence: 99%

“…However, at this stage, a Sabidussi-like theorem concerning GRRs of a prescribed valency is far out of reach. Even in the special case that the valency is 3, although it has attracted attention from several authors over the last few decades [2,4,9,12,16,19,20,21,22], little is yet known of which groups have a cubic GRR.…”

Section: Introductionmentioning

confidence: 99%