Perkel Graph


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 {6,5,2;1,1,3} (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 (-3)^(20)(1/2(3-sqrt(5)))^(18)(1/2(3+sqrt(5)))^(18)6^1.

It is implemented in the Wolfram Language as GraphData["PerkelGraph"].

See also


Explore with Wolfram|Alpha


Brouwer, A. E. "Perkel Graph.", A. E.; Cohen, A. M.; and Neumaier, A. "The Perkel Graph for L(2,19)." §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, "Perkel Graph.", 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 J_1 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.

Cite this as:

Weisstein, Eric W. "Perkel Graph." From MathWorld--A Wolfram Web Resource.

Subject classifications