"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 drawings in LCF
notation of order 6, more than 203 of order 3, and more than 600 of order 2.
Drawings are shown above for LCF notation drawings
of order 6 and bilaterally symmetric drawings of order 3.
It is a subgraph of the Hoffman-Singleton graph obtainable by choosing any edge then deleting the 14 vertices at graph distance at most 1 from an endpoint 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. The Sylvester graph has graph genus 10 (E. Weisstein, Jan. 13, 2026).
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 drawings found from different initial drawings by minimizing the sum of square deviations from unit edge lengths until a local minimum was reached are illustrated above.
The Sylvester configuration graph of a configuration is the set of ordinary points and ordinary lines.
