On the asymptotic enumeration of Cayley graphs

Joy Morris,Mariapia Moscatiello,Pablo Spiga

doi:10.1007/s10231-021-01163-w

Abstract

In this paper, we are interested in the asymptotic enumeration of Cayley graphs. It has previously been shown that almost every Cayley digraph has the smallest possible automorphism group: that is, it is a digraphical regular representation (DRR). In this paper, we approach the corresponding question for undirected Cayley graphs. The situation is complicated by the fact that there are two infinite families of groups that do not admit any graphical regular representation (GRR). The strategy for digraphs involved analysing separately the cases where the regular group R has a nontrivial proper normal subgroup N with the property that the automorphism group of the digraph fixes each N-coset setwise, and the cases where it does not. In this paper, we deal with undirected graphs in the case where the regular group has such a nontrivial proper normal subgroup.

Highlights

We consider only finite groups and finitegraphs in this paper
We say that Γ = (V, E) is a graph if E = {(w, v) ∣ (v, w) ∈ E}, that is, if Γ is a symmetric binary relation
A second piece of the proof involved considering the possibility that the group R has a proper nontrivial normal subgroup N, and there is a digraph automorphism that fixes every orbit of N setwise

Summary

Introduction

We consider only finite groups and finite (di)graphs in this paper. A digraph Γ is an ordered pair (V, E) with V a finite non-empty set of vertices and with E a subset of the Cartesian product V × V. A second piece of the proof involved considering the possibility that the group R has a proper nontrivial normal subgroup N, and there is a digraph automorphism that fixes every orbit of N setwise. Rather than trying to accomplish the full result in a single paper, it makes sense to divide the work into the main pieces that were used to prove the DRR result and attempt to show each of these pieces for GRRs. The first piece, showing that there are not many Cayley graphs admitting graph automorphisms that are group automorphisms (unless the group is generalised dicyclic or abelian of exponent greater than 2) was accomplished by the third author in [22]. Prior to launching into the pieces of the proof mentioned above, we provide some additional background and introductory material

General notation

Graph automorphisms that fix or invert every group element

Groups with a “large” normal subgroup

Groups with a “small” normal subgroup

Specific notation