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SUMMARY:On the existence of certain generalized quadrangles - Felix Lazebn
 ik (University of Delaware)
DTSTART:20080515T133000Z
DTEND:20080515T143000Z
UID:TALK12059@talks.cam.ac.uk
CONTACT:Andrew Thomason
DESCRIPTION:Let s>=2 be an integer.  A generalized triangle of order s is 
 a bipartite (s+1)-regular graph of girth six and diameter three. It is als
 o known as the point-line incidence graph of a finite projective plane of 
 order s\,  and it is known to exist for all s which are prime powers. For 
 each prime power s >= 9 which  is not a prime\, there exist at least two p
 airwise nonisomorphic generalized triangles or order s\, and the number of
  isomorphism classes grows fast with s.\n\nA generalized quadrangle of ord
 er s is a bipartite (s+1)-regular graph of girth eight and diameter four. 
 It is known to exist for all s which are prime powers. Contrary to the cas
 e of generalized triangles\, for each odd prime power s\, only one (up to 
 isomorphism) generalized quadrangle of order s is known.\n\nWe survey some
  recent results on generalized quadrangles\, and study a relation between 
 the existence of certain algebraically defined s-regular graphs of girth e
 ight and diameter six and the existence of generalized quadrangles of orde
 r s. Some questions are reduced to analysis of certain algebraic varieties
  over finite fields and\npermutation polynomials.  We will present some no
 n-existence results and mention several open questions.\n
LOCATION:MR12
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