## Pseudodual grids and extensions of generalized quadrangles.(Russian, English)Zbl 1014.05014

Sib. Mat. Zh. 42, No. 5, 1117-1124 (2001); translation in Sib. Math. J. 42, No. 5, 936-941 (2001).
Let $$\text{GQ}(s,t)$$ be the class of all generalized quadrangles of order $$(s,t)$$ (i.e. a geometry consisting of points and lines such that each line comprises $$s+1$$ points, each point lies on $$t+1$$ lines, and for every point $$a$$ not lying on a line $$L$$ there exists a unique line passing through $$a$$ and intersecting $$L$$). The subgraph $$[a]\cap [b]$$ is called a $$\mu$$-subgraph if the vertices $$a$$ and $$b$$ are at distance 2. Let $$\Gamma$$ be an amply regular, locally $$\text{GQ}(s,t)$$ graph, $$t>1$$. If the $$\mu$$-subgraphs of $$\Gamma$$ are pseudodual grids then it is proved that either $$s=t=2$$ and $$\Gamma$$ is a Taylor graph (a 2-antipodal cover of a clique) or $$\Gamma$$ is the only strongly regular, locally $$\text{GQ}(2,4)$$ graph with parameters $$(64,27,10,12)$$ (i.e. contains 64 vertices, is regular of valency 27, each of its edges lies in 10 triangles, and $$[a]\cap [b]$$ contains 12 vertices for every two vertices $$a$$ and $$b$$ that are at distance 2 in it).

### MSC:

 05B25 Combinatorial aspects of finite geometries 51E12 Generalized quadrangles and generalized polygons in finite geometry 05E30 Association schemes, strongly regular graphs
Full Text: