The Gray graph is a cubic semisymmetric graph on 54 vertices. It was discovered by Marion C. Gray in 1932, and was first published by Bouwer (1968). Malnič et al. (2002) showed that the Gray graph is indeed the smallest possible cubic semisymmetric graph. It is illustrated above in a number of embeddings.
It is the Levi graph of the Gray configuration.
The Gray graph is implemented in the Wolfram Language as GraphData["GrayGraph"].
The Gray graph has a single order-9 LCF notation ![[-25,7,-7,13,-13,25]^9](/images/equations/GrayGraph/Inline1.svg)
and five distinct
order-1 LCF notations.
The Gray graph has graph genus 7 (Marušič et al. 2005, Conder and Stokes 2019, Brinkmann 2020, Metzger and Ulrigg 2025),
girth 8, graph diameter
6, automorphism group order 
, and is the Levi graph
of two dual, triangle-free, point-, line-, and flag-transitive, non-self-dual 
configurations
(Marušič and Pisanski 2000). The symmetric embedding illustrated above
is due to (Marušič and Pisanski 2000).
The Gray graph has graph spectrum
 |
The Gray graph can be constructed by taking three copies of the complete bipartite graph 
and, for a particular edge 
,
subdividing 
in each of the three copies, joining the resulting three vertices to a new vertex,
and repeating with each edge.
