Independent Sets In Association Schemes

Let X be k-regular graph on v vertices and let τ denote the least eigenvalue of its adjacency matrix A(X). If α(X) denotes the maximum size of an independent set in X, we have the following well known bound: α(X) ≤ v 1− k τ . It is less well known that if equality holds here and S is a maximum independent set in X with characteristic vector x, then the vector