Generalized Quadrangle


A generalized quadrangle is a generalized polygon of order 4.

An order-(s,t) generalized quadrangle contains s+1 points in each line and has t+1 lines through every point, giving (s+1)(st+1) points and (t+1)(st+1) lines.

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

graphother namesVgraph spectrum
GQ(2, 1)rook graph K_3 square K_39(-2)^41^44^1
GQ(2, 2)Kneser graph K(6,2), doily of Payne15(-3)^51^96^1
GQ(2, 4)Schläfli graph complement27(-5)^61^(20)10^1
GQ(3, 9)O_6^-(3)112(-10)^(21)2^(90)30^1

The generalized quadrangle GQ(2,1) is the line graph of the complete bipartite graph K_(3,3). 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 Z_3×Z_3. The Goddard-Henning graph can be obtained from GQ(2,1) by removing two edges.


The generalized quadrangle GQ(2,2), commonly denoted W_2, 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 W_2 is the Tutte 8-cage.

The two graphs on 27 vertices obtained by subtraction the spread from GQ(2,4) are distance-regular with intersection array {8,6,1;1,3,8}. One of them is also distance-transitive ( These graphs are cospectral integral graphs with graph spectrum (-4)^6(-1)^82^(12)8^1.

There is a unique generalized quadrangle GQ(3,9), denoted O_6^-(3) (and apparently also U_4(3), 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 GQ(9,3) on 280 points defined by U_4(3), adjacent when they meet) by Brouwer, and this graph is determined by spectrum (van Dam and Haemers 2003). GQ(3,9) is also the first subconstituent of the McLaughlin graph (cf. The local GQ(3,9) graph is known as the Brouwer-Haemers graph. GQ(3,9) has a split into two Gewirtz graphs.

See also

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

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Brouwer, A. E. "The O_6^-(3) Graph.", A. E. "The Sp(4,2) Generalized Quadrangle.", 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, "1st Subconstituent of McLaughlin Graph." "GQ(2,4) Minus Spread.", 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. Dam, E. R. and Haemers, W. H. "Which Graphs Are Determined by Their Spectrum?" Lin. Algebra Appl. 373, 139-162, 2003.

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Generalized Quadrangle

Cite this as:

Weisstein, Eric W. "Generalized Quadrangle." From MathWorld--A Wolfram Web Resource.

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