Perkel graph

Perkel graph
Perkel graphs with 19-fold symmetry
Vertices	57
Edges	171
Radius	3
Diameter	3
Girth	5
Automorphisms	3420
Chromatic number	3
Properties	Regular, distance-transitive
Table of graphs and parameters

In mathematics, the Perkel graph, named after Manley Perkel, is a 6-regular graph with 57 vertices and 171 edges. It is the unique distance-regular graph with intersection array (6, 5, 2; 1, 1, 3). ^[1] The Perkel graph is also distance-transitive.

It is also the skeleton of an abstract regular polytope, the 57-cell.

The vertex set is Z₃× Z₁₉ where (i,j) is joined to (i+1,k) when (k−j)³ = 2⁶ⁱ.

References

1. Coolsaet, K. and Degraer, J. "A Computer Assisted Proof of the Uniqueness of the Perkel Graph." Designs, Codes and Crypt. 34, 155–171, 2005.

• Brouwer, A. E. Perkel Graph. .
• 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.
• Perkel, M. Bounding the Valency of Polygonal Graphs with Odd Girth. Can. J. Math. 31, 1307-1321, 1979.
• Perkel, M. Characterization of in Terms of Its Geometry.Geom. Dedicata 9, 291-298, 1980.

External links

• Weisstein, Eric W. "Perkel graph". MathWorld .