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Sylvester Graph

SylvesterGraph

"The" Sylvester graph is a quintic graph on 36 nodes and 90 edges that is the unique distance-regular graph with intersection array {5,4,2;1,1,4} (Brouwer et al. 1989, §13.1.2; Brouwer and Haemers 1993). It is a subgraph of the Hoffman-Singleton graph obtainable by choosing any edge, then deleting the 14 vertices within distance 2 of that edge.

It has graph diameter 3, girth 5, graph radius 3, is Hamiltonian, and nonplanar. It has chromatic number 4, edge connectivity 5, vertex connectivity 5, and edge chromatic number 5.

It is an integral graph and has graph spectrum 5^12^(16)(-1)^(10)(-3)^9 (Brouwer and Haemers 1993).

The Sylvester graph of a configuration is the set of ordinary points and ordinary lines.

See also

Distance-Regular Graph, Integral Graph, Ordinary Line, Ordinary Point

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References

Brouwer, A. E. "Sylvester Graph." http://www.win.tue.nl/~aeb/drg/graphs/Sylvester.html.Brouwer, A. E.; Cohen, A. M.; and Neumaier, A. §13.1.2 in Distance Regular Graphs. New York: Springer-Verlag, 1989.Brouwer, A. E. and Haemers, W. H. "The Gewirtz Graph: An Exercise in the Theory of Graph Spectra." European J. Combin. 14, 397-407, 1993.DistanceRegular.org. "Sylvester Graph." http://www.distanceregular.org/graphs/sylvester.html.Guy, R. K. "Monthly Unsolved Problems, 1969-1987." Amer. Math. Monthly 94, 961-970, 1987.Guy, R. K. "Unsolved Problems Come of Age." Amer. Math. Monthly 96, 903-909, 1989.

Cite this as:

Weisstein, Eric W. "Sylvester Graph." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/SylvesterGraph.html

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