8 research outputs found
On collineations and dualities of finite generalized polygons
In this paper we generalize a result of Benson to all finite generalized polygons. In particular, given a collineation theta of a finite generalized polygon S, we obtain a relation between the parameters of S and, for various natural numbers i, the number of points x which are mapped to a point at distance i from x by theta. As a special case we consider generalized 2n-gons of order (1,t) and determine, in the generic case, the exact number of absolute points of a given duality of the underlying generalized n-gon of order t
Domesticity in generalized quadrangles
An automorphism of a generalized quadrangle is called domestic if it maps no chamber, which is here an incident point-line pair, to an opposite chamber. We call it point-domestic if it maps no point to an opposite one and line-domestic if it maps no line to an opposite one. It is clear that a duality in a generalized quadrangle is always point-domestic and linedomestic. In this paper, we classify all domestic automorphisms of generalized quadrangles. Besides three exceptional cases occurring in the small quadrangles with orders (2, 2), (2, 4), and (3, 5), all domestic collineations are either point-domestic or line-domestic. Up to duality, they fall into one of three classes: Either they are central collineations, or they fix an ovoid, or they fix a large full subquadrangle. Remarkably, the three exceptional domestic collineatons in the small quadrangles mentioned above all have order 4
Distance-regular Cayley graphs with small valency
We consider the problem of which distance-regular graphs with small valency
are Cayley graphs. We determine the distance-regular Cayley graphs with valency
at most , the Cayley graphs among the distance-regular graphs with known
putative intersection arrays for valency , and the Cayley graphs among all
distance-regular graphs with girth and valency or . We obtain that
the incidence graphs of Desarguesian affine planes minus a parallel class of
lines are Cayley graphs. We show that the incidence graphs of the known
generalized hexagons are not Cayley graphs, and neither are some other
distance-regular graphs that come from small generalized quadrangles or
hexagons. Among some ``exceptional'' distance-regular graphs with small
valency, we find that the Armanios-Wells graph and the Klein graph are Cayley
graphs.Comment: 19 pages, 4 table
Generalised quadrangles with a group of automorphisms acting primitively on points and lines
We show that if G is a group of automorphisms of a thick finite generalised
quadrangle Q acting primitively on both the points and lines of Q, then G is
almost simple. Moreover, if G is also flag-transitive then G is of Lie type.Comment: 20 page