Strongly regular graphs and designs with three intersection numbers (Q5926317)

Let \(D\) be a 2-design with point set \(V\), block set \({\mathcal B}\) and intersection numbers \(\rho_1\), \(\rho_2\), \(\rho_3\) \((\rho_1>\rho_2)\). Suppose blocks in \({\mathcal B}\) are partitioned \({\mathcal B}= {\mathcal B}_1\cup{\mathcal B}_2\cup\cdots\cup{\mathcal B}_s\) so that for distinct blocks \(X\), \(Y\) in \({\mathcal B}\) we have \[ |X\cap Y|= \begin{cases} \rho_3\quad\text{if }X\in B_i,\;Y\in{\mathcal B}_j,\;i\neq j\\ \rho_1\text{ or }\rho_2\quad\text{if }X,Y\in{\mathcal B}_i,\;i= 1,2,\dots,s.\end{cases} \] Asume further that \(\rho_1\) and \(\rho_2\) are realizable in at least one \({\mathcal B}_i\). Then \(D\) is said to admit a nearly affine decomposition. The authors prove that if a 2-design \(D= (V,{\mathcal B})\) with intersection numbers \(\rho_1\), \(\rho_2\), \(\rho_3\) \((\rho_1> \rho_2)\) admits a nearly affine decomposition \({\mathcal B}= {\mathcal B}_1\cup\cdots\cup{\mathcal B}_s\), then graphs \(\Gamma_i\) with vertex set \({\mathcal B}_i\) and edges \(X,Y\in{\mathcal B}_i\), \(X\neq Y\), adjacent iff \(|X\cap Y|= \rho_1\), are all strongly regular graphs, \(1\leq i\leq s\). Moreover, if \(\rho_1\), \(\rho_2\), \(\rho_3\) are distinct, then \(D\) is block schematic (i.e. blocks of \(D\) form an association scheme under a certain equivalence relation defined on blocks). The authors also obtain some results on \(\alpha\)-resolvable designs (a 2-design is \(\alpha\)-resolvable iff its blocks can be partitioned into classes such that each point of \(D\) occurs a constant number \(\alpha\) times amongst the blocks of each class). A family of \(\alpha\)-resolvable designs with at most three intersection numbers and admitting a nearly affine decomposition is constructed, too.