New strongly regular graphs on open parameter sets of Brouwer's table

We construct strongly regular graphs for nine parameter sets that were marked as open in Brouwer's table: one srg(250,81,24,27), two srg(300,69,18,15), two srg(486,100,22,20), two srg(486,97,16,20), and, as lower bounds on the number of pairwise non-isomorphic graphs, 35 vertex-transitive srg(320,87,22,24), 95 vertex-transitive srg(320,88,24,24), 22 srg(324,114,36,42), 10 srg(324,133,52,56) and 93 srg(324,136,58,56). All nine sets are now marked as realised in the table, on the basis of this work. For the two sets on 320 vertices, existence already follows from a known abelian difference set by a sum-graph construction; our graphs are, to our knowledge, the first vertex-transitive ones. Most graphs were found by computer searches over unions of orbitals of coset actions, the others by exact searches for partial difference sets. Many are Cayley graphs; they give partial difference sets in nonabelian groups of orders 320 and 486 and reversible (320,88,24) difference sets, which do not exist in abelian groups. For srg(250,81,24,27), srg(300,69,18,15) and one cyclotomic srg(324,136,58,56) on F_4 × F_81 we prove strong regularity by hand. Every graph is checked by an exact computation, and the graphs are distributed in graph6 format with a short verifier.