In mathematics, the Perkel graph, named after Manley Perkel, is a 6-
regular graph
In graph theory, a regular graph is a graph where each vertex has the same number of neighbors; i.e. every vertex has the same degree or valency. A regular directed graph must also satisfy the stronger condition that the indegree and outdegree o ...
with 57 vertices and 171 edges. It is the unique
distance-regular graph
In the mathematical field of graph theory, a distance-regular graph is a regular graph such that for any two vertices and , the number of vertices at distance from and at distance from depends only upon , , and the distance between and .
...
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
A skeleton is the structural frame that supports the body of an animal. There are several types of skeletons, including the exoskeleton, which is the stable outer shell of an organism, the endoskeleton, which forms the support structure inside ...
of an
abstract regular polytope
In mathematics, an abstract polytope is an algebraic partially ordered set which captures the dyadic property of a traditional polytope without specifying purely geometric properties such as points and lines.
A geometric polytope is said to be ...
, the
57-cell
In mathematics, the 57-cell (pentacontakaiheptachoron) is a self-dual abstract regular 4-polytope ( four-dimensional polytope). Its 57 cells are hemi-dodecahedra. It also has 57 vertices, 171 edges and 171 two-dimensional faces.
The symmetry or ...
.
References
* 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
* {{Mathworld , urlname = PerkelGraph , title = Perkel graph
Individual graphs
Regular graphs