 TOPICS   A generalized quadrangle is a generalized polygon of order 4.

An order- generalized quadrangle contains points in each line and has lines through every point, giving points and lines.

The following table summarizes the vertex counts and spectra of some generalized quadrilaterals.

 graph other names graph spectrum GQ(2, 1) rook graph 9 GQ(2, 2) Kneser graph , doily of Payne 15 GQ(2, 4) Schläfli graph complement 27 GQ(3, 9) 112 The generalized quadrangle is the line graph of the complete bipartite graph . It is also the (2, 3)-Hamming graph, (3, 3)-rook graph, (3, 3)-rook complement graph, 9-Paley graph, and quartic vertex-transitive graph Qt9. It is also a conference graph (Godsil and Royle 2001, p. 222), as well as the Cayley graph of the Abelian group . The Goddard-Henning graph can be obtained from by removing two edges. The generalized quadrangle , commonly denoted , is illustrated above. It is also the (6,2)-Kneser graph and is also known as the doily of Payne (Payne 1973). It can be constructed by dividing six points into three pairs in all fifteen different ways, then connecting sets with common pairs (hence its isomorphism with a Kneser graph). The Levi graph of is the Tutte 8-cage.

The two graphs on 27 vertices obtained by subtraction the spread from are distance-regular with intersection array . One of them is also distance-transitive (DistanceRegular.org). These graphs are cospectral integral graphs with graph spectrum .

There is a unique generalized quadrangle , denoted (and apparently also , though this notation seems to refer to the fact that it may be described as the graph on the 112 totally isotropic lines of the on 280 points defined by , adjacent when they meet) by Brouwer, and this graph is determined by spectrum (van Dam and Haemers 2003). is also the first subconstituent of the McLaughlin graph (cf. DistanceRegular.org). The local graph is known as the Brouwer-Haemers graph. has a split into two Gewirtz graphs.

Brouwer-Haemers Graph, Generalized Dodecagon, Generalized Hexagon, Generalized Octagon, Generalized Polygon, Tutte 8-Cage

## Explore with Wolfram|Alpha More things to try:

## References

Brouwer, A. E. "The Graph." http://www.win.tue.nl/~aeb/drg/graphs/U4_3.html.Brouwer, A. E. "The Sp(4,2) Generalized Quadrangle." http://www.win.tue.nl/~aeb/drg/graphs/GQ22.html.Brouwer, A. E.; Cohen, A. M.; and Neumaier, A. "Generalized Quadrangles with Line Size Three." §1.15 in Distance Regular Graphs. New York: Springer-Verlag, pp. 29-33, 1989.Cameron, P. J.; Goethals, J. M.; and Seidel, J. J. "Strongly Regular Graph having Strongly Regular Subconstituents." J. Algebra 55, 257-280, 1978.DistanceRegular.org. "1st Subconstituent of McLaughlin Graph." http://www.distanceregular.org/graphs/1sub-mclaughlingraph.html.DistanceRegular.org. " Minus Spread." http://www.distanceregular.org/graphs/gq2.4minusspread.html.Godsil, C. and Royle, G. "Generalized Quadrangles." §10.8 in Algebraic Graph Theory. New York: Springer-Verlag, pp. 235-237, 2001.Payne, S. E. "Finite Generalized Quadrangles: A Survey." Proceedings of the International Conference on Projective Planes. Washington State Univ. Press, pp. 219-261, 1973.Polster, B. "Pretty Pictures of Geometries." Bull. Belg. Math. Soc. 5, 417-425, 1998. http://projecteuclid.org/Dienst/UI/1.0/Summarize/euclid.bbms/1103409021/.van Dam, E. R. and Haemers, W. H. "Which Graphs Are Determined by Their Spectrum?" Lin. Algebra Appl. 373, 139-162, 2003.