However, care must be taken with this definition since arc-transitive or a 1-arc-transitive graphs are sometimes also known as symmetric graphs (Godsil
and Royle 2001, p. 59). This can be especially confusing given that there exist
graphs that are symmetric in the sense of vertex-
and edge-transitive, but not arc-transitive.
In other words, graphs exist for which any edge can be mapped to any other--but in
only one of the two possible ways. Such graphs were first considered by Tutte (1966),
who showed that any such graph must be regular of
even degree. The first examples were given by Bouwer (1970), whose smallest example
had 54 vertices was quartic. Dolye (1976) and Holt
(1981) subsequently and independently discovered a beautiful quartic
symmetric graph on 27 vertices, known as the Doyle
graph (or sometimes the Holt graph), that is not arc-transitive.
This graph can be obtained from Bouwer's 54-vertex graph by identifying pairs of
diametrically opposed vertices (Doyle 1998).

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