Bouwer graphs, a term coined here for the first time, are a family of regular graphs which includes members that are half-arc-transitive, meaning both edge-transitive and vertex-transitive but not arc-transitive.
Bouwer's general construction defines a graph 
with 
and 
such that 
. The vertex set
of this graph is identified with the Cartesian product
 |
where 
denotes the ring of integers modulo 
, and the edge set consists of
pairs of 
-tuples
 |
for 
, ..., 
(with addition mod 
) and 
, ..., 
such that either 
for all 
, 3, ..., 
, or else there is exactly one 
for which 
, in which case it is taken as 
(mod 
).
Bouwer graphs are symmetric by construction, and include the following named graphs which are arc-transitive.
 | graph |
 | cycle graph  |
 | generalized hexagon GH(2,1) |
 | circulant
graph  |
 | 525-Haar graph |
 | quartic vertex-transitive graph Qt66 |
 | Pappus graph |
Not all Bouwer graphs are arc-transitive. Bouwer (1970) gave the first examples of half-arc-transitive
graphs, showing that 
is a connected 
-regular half-arc-transitive
graph for all integers 
. This class of graphs has 
vertices, giving graphs with vertex counts 54,
486, 4374, 39366, 354294, ... for 
, 3, ....
Conder and Žitnik (2016) subsequently proved that almost all Bouwer graphs are half-arc-transitive. In particular,
is half-arc-transitive
whenever 
and 
.
This smallest 
example is the quartic symmetric graph
on 54 vertices illustrated above in several embeddings. It is half-arc-transitive
and can be concisely described and constructed from the vertex
set 
,
where 
is joined to 
,
, and 
(Holt 1981).
Doyle (1976) and Holt (1981) independently discovered the smaller half-arc-transitive graph now known as the Doyle graph, which can be obtained from the 54-vertex Bouwer graph by identifying pairs of diametrically opposed vertices (Doyle 1998).
A partial tabulation of small half-arc-transitive graphs constructed using Bouwer's method is given in the following table (Weisstein,
Nov. 17, 2010), where 
is the vertex count. These
graphs are implemented in the Wolfram
Language as GraphData[
"Bouwer", 
N, m, n
].
 |  |
| 54 |  |
| 60 |  |
| 63 |  |
| 84 |  |
| 100 |  |
