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 = p h , 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 < q 1 / 6 / 8 ). Both algebraic and combinatorial tools are used.