Conference matrices from projective planes of order 9
The four known nonisomorphic projective planes of order 9 determine seven nonismorphic affine planes of order 9. From the corresponding transversal designs T(10,9), one finds \(7\times \left( \begin{matrix} 10\\ 5\end{matrix} \right)=1764\) transversal designs T(5,9). Each of these defines a strongly regular graph with parameters (81,40,19,20), i.e. a so-called half-case s.r.g., and a corresponding conference matrix. It is shown that among these there are 175 nonisomorphic strongly regular graphs and 26 nonequivalent conference matrices. These matrices are given in tables. These results depend on a computer search combined with symmetry arguments and the following theorem. Theorem: If K is a graph on four vertices and G is a half-case s.r.g., then G contains as many copies of K as of its complement.
scientific article; zbMATH DE number 3906507