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Perkel graph

6-regular graph with 57 vertices and 171 edges

6-regular graph with 57 vertices and 171 edges

FieldValue
namePerkel graph
image[[Image:Perkel graph embeddings.svg320px]]
image_captionPerkel graphs with 19-fold symmetry
vertices57
edges171
automorphisms3420
radius3
diameter3
girth5
chromatic_number3
propertiesRegular, 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). 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

  • 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.

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.
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This article was imported from Wikipedia and is available under the Creative Commons Attribution-ShareAlike 4.0 License. Content has been adapted to SurfDoc format. Original contributors can be found on the article history page.

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