Generalized Quadrangle

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.