Semi-regular graph automorphisms and generalized quadrangles

Automorphisms and characterizations of finite generalized quadrangles

Our paper surveys some new developments in the theory of automorphisms and characterizations of finite generalized quadrangles. It is the purpose to mention important new results which did not appear in the following standard works (or surveys) on the subject: Collineations of finite generalized quadrangles (S. E. Payne, 1983), Finite Generalized Quadrangles (S. E. Payne and J. A. Thas, 1984), ...

Elation generalized quadrangles with extra automorphisms and trivial spans

We show that the parameters of a finite elation generalized quadrangle for which the point spans containing the elation point (∞) are trivial, and which admit extra automorphisms fixing (∞) linewise, are powers of the same prime, so that the elation group is a p-group for some prime p. This observation has several strong corollaries, perhaps the most remarkable of which is that a thick elation ...

Split Semi-Biplanes in Antiregular Generalized Quadrangles

There are a number of important substructures associated with sets of points of antiregular quadrangles. Inspired by a construction of P. Wild, we associate with any four distinct collinear points p, q, r and s of an antiregular quadrangle an incidence structure which is the union of the two biaffine planes associated with {p, r} and {q, s}. We investigate when this incidence structure is a sem...

Finite Generalized Quadrangles

Monomial graphs and generalized quadrangles

Let Fq be a finite field, where q = p for some odd prime p and integer e ≥ 1. Let f, g ∈ Fq[x, y] be monomials. The monomial graph Gq(f, g) is a bipartite graph with vertex partition P ∪L, P = Fq = L, and (x1, x2, x3) ∈ P is adjacent to [y1, y2, y3] ∈ L if and only if x2 + y2 = f(x1, y1) and x3 + y3 = g(x1, y1). Dmytrenko, Lazebnik, and Williford proved in [6] that if p ≥ 5 and e = 23 for integ...