(v,k, λ)-Graphs and polarities of (v,k, λ)-designs

In 2.1 it is established that there is a one-to-one correspondence between (v, k, )-graphs and polarities, with no absolute points, of (v, k, )-designs. This is used to show that the parameters of a (v, k, )-graph are of the form ((s/a)((s + a)²–1), s(s+a), sa) where s and a are positive integers with a dividing s(s²–1) (Theorem 3.4) but strictly less than s(s²–1) (Proposition 4.3). Some consequences of this parametrization are discussed and in particular, it is shown that for fixed 2 there are only finitely many non-isomorphic (v, k, )-graphs. In 4. it is shown that (v, k, )-graphs can also be constructed using polarities, with all points absolute, of certain designs. In 5. isomorphisms and automorphisms of graphs and designs are discussed. Many examples of (v, k, )-graphs, including some apparently new ones, are given.Dedicated to Peter Dembowski, 28 January 1971