Marston Conder (University of Auckland, New Zealand)

Title: Some new development on edge-transitive graphs

Abstract:

Vertex-transitive graphs (and the sub-class of Cayley graphs) have 
been the object of study for many decades.
On the other hand, there are many important graphs that are edge-transitive 
but not vertex-transitive (such as $K_{m,n}$ with $m e n$), 
and many vertex-transitive graphs that are not Cayley graphs  
(such as the Petersen graph, the Gray graph, and the Hoffman-Singleton graph).  

In this talk, I will describe some recent developments in the theory and
techniques for construction of edge-transitive  graphs of various kinds. 

First, I will briefly describe a new method of finding all edge-transitive 
graphs of up to given (small) order, which has resulted in quickly finding  
all such graphs of order less than $48$.  This is joint work with Gabriel Verret. 

Then for the rest of the talk I will summarise some recent work 
with Yanquan Feng, Mi-Mi Zhang & Jinxin Zhou (Beijing) on {em bi-Cayley graphs}, 
which are graphs that admit a group $H$ of automorphisms acting semi-regularly 
on the vertices, with two orbits (of the same length).  These include the 
Petersen graph and the Gray graph, and many more besides.

Our main focus was on the case where the `bi-Cayley' group $H$ is normal 
in the full automorphism group of the graph, and produced infinite families 
of examples in each of three sub-classes of bi-Cayley graphs, namely 
those that are arc-transitive, half-arc-transitive or semisymmetric, 
respectively.

In doing this, we found the answer to a number of open questions  
about these and related classes of graphs, posed by Li (in 
{em Proc. American Math. Soc.} 133 (2005)), Marušič and 
Potočnik (in {em European J. Combinatorics/} 22 (2001)) and 
Marušič and Šparl (in {em J. Algebraic Combinatorics/} 28 (2008)).  
Also we found and corrected an error in a recent paper by Li, Song 
and Wang (in {em J. Combinatorial Theory, Series A} 120 (2013)).