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Perkel Graph

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PerkelGraph

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

57-Cell

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References

Brouwer, A. E. "Perkel Graph." http://www.win.tue.nl/~aeb/graphs/Perkel.html.Brouwer, 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, 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 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. https://mathworld.wolfram.com/PerkelGraph.html

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