On classification of two class partially balanced designs (Q5935439)

Consider a regular graph \(G= (V,E)\) where \(V\) and \(E\) denote respectively the set of vertices and the set of edges. Let \(v\) be the number of elements in \(V\), and \(k\), \(\lambda\), \(\mu\) be integers \(\geq 0\) with \(2\leq k\leq v-1\) and \(\lambda\neq \mu\). A \((v,k,\lambda,\mu)\)-design over \(G\) is a pair \(D= (V,B)\) where \(B\) is a set of \(k\)-subsets of \(V\), called blocks, such that if \(i\) and \(j\) are any distinct members of \(V\), then there are exactly \(\lambda\) blocks containing \(\{i,j\}\) if \(\{i,j)\in E\) and exactly \(\mu\) blocks containing \(\{i,j\}\) if \(\{i,j\}\not\in E\). For such designs, the replication number \(r\) is at least \(2\lambda-\mu\). In this paper the authors are concerned with designs for which \(r= 2\lambda- \mu\), and they present a complete classification of such designs in case \(G\) has eigenvalue \(-2\). It is shown that there are three infinite families (and their complements) and numerous sporadic designs. The sporadic designs correspond to the Peterson, Clebsch, and Shrikhande graphs and the triangular graph \(T(4)\).