Strongly regular graphs with strongly regular decomposition

Abstract

The title refers to strongly regular graphs r, which admit a partition { X,, X,} of the vertex set such that each of the induced subgraphs r1 and l?, on X, and X, respectively is strongly regular, a clique, or a coclique. A central role is played by the design D having point set X,, block set X,, and incidence given by adjacency in r,. If rl is a clique or a coclique and r,, is primitive, D must be a quasisymmetric design. If rl and r, are both strongly regular, D is a strongly regular design in the sense of D. G. Higman [14], except possibly when r, is the graph of a regular conference matrix. Conversely, a quasisymmetric or strongly regular design with suitable parameters gives rise to a strongly regular graph with strongly regular decomposition. Moreover, if r, and rl are strongly regular with suitable parameters, then r, must be strongly regular too. We give several examples and some nonexistence results. We include a table of all feasible parameter sets up to 300 vertices. For most of the cases in the table existence or nonexistence is settled. Some of the results in this paper are old, due to M. S. Shrikhande [17], W. G. Bridges and M. S. Shrikhande [3], and W. H. Haemers [13]. We mainly use eigenvalue techniques. We need results on interlacing eigenvalues (see [13]). Two sequences p1 > . . . 2 p,, and u1 > . . . 2 a,