Perkel graph
{{Short description|6-regular graph with 57 vertices and 171 edges}}
{{Infobox graph
| name = Perkel graph
| image = 320px
| image_caption = Perkel graphs with 19-fold symmetry
| namesake =
| vertices = 57
| edges = 171
| automorphisms = 3420
| radius = 3
| diameter = 3
| girth = 5
| chromatic_number = 3
| chromatic_index =
| fractional_chromatic_index =
| properties = Regular, distance-transitive
}}
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).Coolsaet, K. and Degraer, J. "A Computer Assisted Proof of the Uniqueness of the Perkel Graph." Designs, Codes and Crypt. 34, 155–171, 2005. The Perkel graph is also distance-transitive.
It is also the skeleton of an abstract regular polytope, the 57-cell.
The vertex set is {{math|Z3 × Z19}} where {{math|(i,j)}} is joined to {{math|(i+1,k)}} when {{math|(k−j)3}} = {{math|26i}}.
References
{{reflist}}
- 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.
- 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
- {{Mathworld | urlname = PerkelGraph | title = Perkel graph}}