Bouwer graphs, a term coined here for the first time, are a family of regular graphs which includes members that are symmetric but not arc-transitive. Such graphs are termed 1/2-transitive by Alspach et al. (1994).
Bouwer's general construction of such graphs 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 
).
Such 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 |
However, this class of graphs also includes members that are symmetric not not edge-transitive. Such graphs
were first considered by Tutte (1966), who did not construct any, but showed that
if it existed, any such graph must be regular of
even degree. The first examples were therefore given by Bouwer (1970), who showed
is a connected 
-regular symmetric arc-intransitive graph for all integers
. This class of graphs has 
vertices, giving graphs with
vertex counts 54, 486, 4374, 39366, 354294, ... for 
, 3, ....
This smallest 
example of such a graph is the quartic symmetric
graph on 54 vertices illustrated above in several embeddings. This graph can
be concisely described and constructed from the vertex
set 
,
where 
is joined to 
,
, and 
(Holt 1981).
Dolye (1976) and Holt (1981) subsequently discovered the smaller symmetric arc-intransitive graph now known as the Doyle graph, which can be obtained from Bouwer's 54-vertex graph by contracting pairs of diametrically opposed vertices (Doyle 1998).
A partial tabulation of small symmetric arc-intransitive graphs constructed using Brouwer'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 |  |
