Menu
Strongly regular graphs are graphs in which every adjacent pair of vertices share \(\lambda\) common neighbours and every non-adjacent pair share \(\mu\) common neighbours. We are interested in strongly regular graphs with \(\lambda = \mu = k\) such that every such set of \(k\) vertices common to any pair always induces a subgraph with a constant number \(x\) of edges. The Friendship Theorem proves that there are no such graphs when \(\lambda = \mu = 1\). We derive constraints which such graphs must satisfy in general, when \(\lambda = \mu > 1\), and \(x \geq 0\), and we find the set of all parameters satisfying the constraints. The result is an infinite, but sparse, collection of parameter sets. The smallest parameter set for which a graph may exist has \(4896\) vertices, with \(k = 1870\).