The Schläfli graph is a strongly regular graph on 27 nodes which is the graph complement
of the generalized quadrangle 
. It is the unique strongly
regular graph with parameters 
(Godsil and Royle 2001, p. 259). It is illustrated
above in two embeddings, the first one corresponding to an order-9 generalized LCF notation.
It is distance-regular with intersection array 
.
It is also distance-transitive.
It is an integral graph with graph spectrum 
.
See also
Generalized Quadrangle,
Strongly Regular Graph
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References
Brouwer, A. E. "Schläfli Graph." http://www.win.tue.nl/~aeb/drg/graphs/Schlaefli.html.Brouwer,
A. E.; Cohen, A. M.; and Neumaier, A. Distance
Regular Graphs. New York: Springer-Verlag, pp. 103-104 and 312, 1989.Brouwer,
A. E. and van Lint, J. H. "Strongly Regular Graphs and Partial Geometries."
In Enumeration
and Design: Papers from the conference on combinatorics held at the University of
Waterloo, Waterloo, Ont., June 14-July 2, 1982 (Ed. D. M. Jackson
and S. A. Vanstone). Toronto, Canada: Academic Press, pp. 85-122,
1984.DistanceRegular.org. "Schläfli Graph." http://www.distanceregular.org/graphs/schlafli.html.Godsil,
C. and Royle, G. Algebraic
Graph Theory. New York: Springer-Verlag, p. 259, 2001.Seidel,
J. J. "Strongly Regular Graphs with 
Adjacency Matrix Having Eigenvalue 3." Lin.
Alg. Appl. 1, 281-298, 1968.
Cite this as:
Weisstein, Eric W. "Schläfli Graph."
From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/SchlaefliGraph.html
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