On the Connectivity of Graphs in Association Schemes

Let (X,R) be a commutative association scheme and let Γ = (X,R ∪R>) be a connected undirected graph where R ∈ R. Godsil (resp., Brouwer) conjectured that the edge connectivity (resp., vertex connectivity) of Γ is equal to its valency. In this paper, we prove that the deletion of the neighborhood of any vertex leaves behind at most one non-singleton component. Two distinct vertices a, b ∈ X are called “twins” in Γ if they have identical neighborhoods: Γ(a) = Γ(b). We characterize twins in polynomial association schemes and show that, in the absence of twins, the deletion of any vertex and its neighbors in Γ results in a connected graph. Using this and other tools, we prove lower bounds on the connectivity of Γ, especially in the case where Γ has diameter two. Among the applications of these results, we prove that the only connected relations in symmetric association schemes which admit a disconnecting set of size two are those which are ordinary polygons.