A strongly regular decomposition of the complete graph and its association scheme

Abstract For any positive integer m, the complete graph on 2 2 m ( 2 m + 2 ) vertices is decomposed into 2 m + 1 commuting strongly regular graphs, which give rise to a symmetric association scheme of class 2 m + 2 − 2 . Furthermore, the eigenmatrices of the symmetric association schemes are determined explicitly. As an application, the eigenmatrix of the commutative strongly regular decomposition obtained from the strongly regular graphs is derived.