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The Perkel graph is a weakly regular graph on 57 vertices and 171 edges, shown above in several embeddings. It is the unique
distance-regular graph with intersection
array 
(Coolsaet and Degraer 2005, Brouwer). The Perkel graph is also distance-transitive.
It is also the skeleton of the 57-cell.
It has graph spectrum 
.
It is implemented in the Wolfram Language as GraphData["PerkelGraph"].
." §13.3 in Distance
Regular Graphs. New York: Springer-Verlag, pp. 401-403, 1989.Coolsaet,
K. and Degraer, J. "A Computer Assisted Proof of the Uniqueness of the Perkel
Graph." Designs, Codes and Crypt. 34, 155-171, 2005.DistanceRegular.org.
"Perkel Graph." http://www.distanceregular.org/graphs/perkel.html.Neumann,
P. M. "Permutationsgruppen von Primzahlgrad und verwandte Themen."
Vorlesungen Math. Inst. Giesen, Heft 5, 1977.Perkel, M. "Bounding
the Valency of Polygonal Graphs with Odd Girth." Canad. J. Math. 31,
1307-1321, 1979.Perkel, M. "Characterization of 
in Terms of Its Geometry." Geom. Dedicata 9,
291-298, 1980.van Dam, E. R. and Haemers, W. H. "Spectral
Characterizations of Some Distance-Regular Graphs." J. Algebraic Combin. 15,
189-202, 2003.
Weisstein, Eric W. "Perkel Graph." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/PerkelGraph.html
