PARTITIONING STRONGLY REGULAR GRAPHS (BALANCED INCOMPLETE BLOCK DESIGNS, ASSOCIATION SCHEMES).

A strongly regular graph can be design partitioned if the vertices of the graph can be partitioned into two sets V and B such that V is a coclique and every vertex in B is adjacent to the same number of vertices in V. In this case, a balanced incomplete block design can be defined by taking elements of V as objects and elements of B as blocks. Many strongly regular graphs can be design partitioned. The nation of design partitioning is extended to a partitioning by a generalization of block designs called order-free designs. All strongly regular graphs can be partitioned via order-free designs. Order-free designs are used to show the nonexistence of a strongly regular graph with parameters (50,28,18,12). The existence of this graph was previously undecided. A computer algorithm that attempts to construct the adjacency matrix of a strongly regular graph (given a suitable order-free design) is presented. Two appendices related to the algorithm are included. The first lists all parameter sets (n,a,c,d) with n ≤ 50 and a ≠ d that satisfy the standard feasibility conditions for strongly regular graphs. Additional information is included for each set. The second appendix contains adjacency matrices (with the partitioning by cocliques and order-free designs exhibited) for most of the parameter sets in the first appendix. The theoretical development is presented in the context of association schemes. Partitioning by order-free designs extends naturally to any association scheme when cocliques are generalized to {Ø,i} -cliques. This extended partitioning is applied to generalized hexagons