The Doyle graph, sometimes also known as the Holt graph (Marušič et al. 2005) or Doyle-Holt graph (Conder and Stokes 2019), is the quartic symmetric graph on 27 nodes illustrated above in several embeddings. It is a half-arc-transitive graph, meaning that it is both edge-transitive and vertex-transitive but not arc-transitive. In other words, any edge of the Doyle graph can be mapped to any other, but in only one of the two possible ways.
The Doyle graph is the unique smallest half-arc-transitive graph (Alspach et al. 1994). Such graphs are also called 1/2-transitive graphs. Note that while Holt (1981) mentions that a referee informed him that Kornya found another example with 27 vertices, the Doyle graph is in fact the only 1/2-transitive graph with 27 vertices and degree 4 (Alspach et al. 1994).
Doyle discovered this graph in 1976, and Holt independently rediscovered it in 1981. It can be obtained from the 54-vertex Bouwer graph
by identifying pairs of diametrically opposed vertices (Doyle 1998). It is also a
subgraph of the 
-Hamming graph. Several embeddings are illustrated above,
the first of which is due to Doyle (1998) and the last due to Marušič
et al. (2005).
It is implemented in the Wolfram Language as GraphData["DoyleGraph"].
The graph can be concisely described and constructed from the vertex set 
,
where 
is joined to 
and 
(Holt 1981).
The Doyle graph is a unit-distance graph. A number of unit-distance embeddings are illustrated above, including edge-vertex degenerate embeddings due to E. Gerbracht (pers. comm., Dec. 27, 2009) and E. Weisstein (Oct. 26, 2023), and a beautiful (and unique) maximally symmetric embedding due to J. Tan (pers. comm., Oct. 16, 2021).
The Doyle graph has two distinct LCF notations of order 9 and sixteen of order 3, illustrated above, together with 1818 of order one.
The Doyle graph has graph genus 5 (Conder and Stokes 2019, Brinkman 2020).
