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
- ^ 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. [1].
- 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.