On regular graphs of girth six arising from projective planes

In 1967, Brown constructed small k-regular graphs of girth six as induced subgraphs of the incidence graph of a projective plane of order q, q≥k. Examining the construction method, we prove that starting from PG(2,q), q=ph, p prime, there are no other constructions using this idea resulting in a (q+1−t)-regular graph of girth six than the known ones, if t is not too large (t≤p and roughly t