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Large Witt Graph

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The large Witt graph, also called the octad graph (Brouwer) or Witt graph (DistanceRegular.org), is the graph whose vertices are the 759 blocks of a Steiner system S(5,8,24) in which two blocks are adjacent whenever they are disjoint (Brouwer et al. 1989, p. 366).

Perhaps the simplest construction is by selecting the 759 codewords of weight 8 of the extended binary Golay code and joining two words when they have disjoint support (i.e., if the codeword vectors are orthogonal).

It is a distance-regular graph with intersection array {30,28,24;1,3,15} and is also distance-transitive. It is an integral graph with graph spectrum (-15)^(23)(-3)^(483)7^(252)30^1. Its automorphism group is the largest Mathieu group M24, of order |M_(24)|=244823040 (Brouwer). Its chromatic number is apparently unknown.

The large Witt graph is implemented in the Wolfram Language as GraphData["LargeWittGraph"].

See also

Doubly Truncated Witt Graph, Golay Code, Ivanov-Ivanov-Faradjev Graph, Mathieu Group M24, Truncated Witt Graph, Witt Design

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References

Brouwer, A. E. "The Octad Graph." https://aeb.win.tue.nl/graphs/M24.html.Brouwer, A. E.; Cohen, A. M.; and Neumaier, A. "The Witt Graph Associated to M_(24)." §11.4A in Distance Regular Graphs. New York: Springer-Verlag, pp. 194, 366-367, and 428, 1989.DistanceRegular.org. "Witt Graph." https://www.math.mun.ca/distanceregular/graphs/witt.html.

Referenced on Wolfram|Alpha

Large Witt Graph

Cite this as:

Weisstein, Eric W. "Large Witt Graph." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/LargeWittGraph.html

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