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.

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.