"The" Sylvester graph is a quintic graph on 36 nodes and 90 edges that is the unique distance-regular
graph with intersection array 
(Brouwer et al. 1989, §13.1.2; Brouwer
and Haemers 1993). It has at least 10 distinct LCF embeddings
of order 6, more than 203 of order 3, and more than 600 of order 2. Embeddings are
shown above for LCF embeddings of order 6 and bilterally symmetric embeddings of
order 3.
It is a subgraph of the Hoffman-Singleton graph obtainable by choosing any edge then deleting the 14 vertices within distance 2 of that edge.
It has graph diameter 3, girth 5, graph radius 3, is Hamiltonian, and nonplanar. It has chromatic number 4, edge connectivity 5, vertex connectivity 5, and edge chromatic number 5.
It is an integral graph and has graph spectrum 
(Brouwer and Haemers 1993).
The Sylvester graph satisfies the rhombus constraints and contains no known unit-distance forbidden subgraph, yet appears not to be a unit-distance. A number of embeddings found from different initial embeddings by minimizing the sum of square deviations from unit edge lengths until a local minimum was reached are illustrated above.
The Sylvester conifguration graph of a configuration is the set of ordinary points and ordinary lines.
