Edge-transitive graphs of small order and the answer to a 1967 question by Folkman

Marston D E Conder,Gabriel Verret

doi:10.5802/alco.82

Abstract

In this paper, we introduce a method for finding all edge-transitive graphs of small order, using faithful representations of transitive permutation groups of small degree, and we explain how we used this method to find all edge-transitive graphs of order up to 47, and all bipartite edge-transitive graphs of order up to 63. We also give an answer to a 1967 question of Folkman about semi-symmetric graphs of large valency; in fact we show that for semi-symmetric graphs of order 2n and valency d, the ratio d/n can be arbitrarily close to 1.

Highlights

Graphs with a large automorphism group hold a significant place in mathematics, dating back to the time of first recognition of the Platonic solids, and in other disciplines where symmetry play an important role, such as structural chemistry, and interconnection networks
A major class of such graphs are the vertex-transitive graphs, whose automorphism group has a single orbit on vertices
Important sub-classes are those of Cayley graphs, and arc-transitive graphs, which include the graphs underlying regular maps on surfaces, and the somewhat less well known class of half-arc-transitive graphs, which are vertex- and edge-transitive but not arc-transitive

Summary

Introduction

Graphs with a large automorphism group hold a significant place in mathematics, dating back to the time of first recognition of the Platonic solids, and in other disciplines where symmetry (and even other properties such as rigidity) play an important role, such as structural chemistry, and interconnection networks. A major class of such graphs are the vertex-transitive graphs, whose automorphism group has a single orbit on vertices. Important sub-classes are those of Cayley graphs (graphs for which some group of automorphisms acts sharply-transitively on vertices), and arc-transitive graphs, which include the graphs underlying regular maps on surfaces, and the somewhat less well known class of half-arc-transitive graphs, which are vertex- and edge-transitive but not arc-transitive. For if X is a vertex-transitive graph of order n with automorphism group A, and ∆ is the neighbourhood of some vertex v, ∆ is preserved by Av and so ∆ is a union of orbits of Av. let G be any transitive group on a set Ω of size n.

Further background

Our approach

Implementation

Summary of our results

Answers to Folkman’s questions