Let c k be the smallest number of vertices in a regular graph with valency k and girth 8. It is known that c k+1 ≥ 2(1+k+k 2 +k 3 ) with equality if and only if there exists a finite generalized quadrangle of order k. No such quadrangle is known when k is not a prime power. In this case, small regular graphs of valency k+1 and girth 8 can be constructed from known generalized quadrangles of order q>k by removing a part of its structure. We investigate the case when q = k+1 is a prime power, and try to determine the smallest graph under consideration that can be constructed from a generalized quadrangle of order q. This problem appears to be much more difficult than expected. We have general bounds and improve these for the classical generalized quadrangle Q(4, q), q even. q

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In [11], weighted {delta(q + 1), delta; k - 1, q}-minihypers, q square, were characterized as a sum of lines and Baer subgeometries PG(3 root q) provided delta is sufficiently small. We extend this result to a new characterization result on weighted {delta upsilon(mu+1),delta upsilon(mu); k - 1, q}-minihypers. We prove that such minihypers are sums of p-dimensional subspaces and of (projected) (2 mu + 1)-dimensional Baer subgeometries