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

TL;DR

This paper presents a method to classify small order edge-transitive graphs using permutation group representations, and answers a 1967 question about semi-symmetric graphs' valency ratios.

Contribution

Introduces a new method for finding all small order edge-transitive graphs and provides an answer to Folkman's 1967 question on semi-symmetric graphs.

Findings

All edge-transitive graphs up to order 47 found

All bipartite edge-transitive graphs up to order 63 identified

Semi-symmetric graphs can have valency-to-order ratio arbitrarily close to 1

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 $1$.