On the connectivity of graphs in association schemes

Abstract

Let $(X,\mathcal{R})$ be a commutative association scheme and let $\Gamma=(X,R\cup R^\top)$ be a connected undirected graph where $R\in \mathcal{R}$. Godsil (resp., Brouwer) conjectured that the edge connectivity (resp., vertex connectivity) of $\Gamma$ 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 vertices $a,b\in X$ are called "twins" in $\Gamma$ if they have identical neighborhoods: $\Gamma(a)=\Gamma(b)$. We characterize twins in polynomial association schemes and show that, in the absence of twins, the deletion of any vertex and its neighbors in $\Gamma$ results in a connected graph. Using this and other tools, we find lower bounds on the connectivity of $\Gamma$, especially in the case where $\Gamma$ has diameter two. Among the applications of these results, we find that the only connected relations in symmetric association schemes which admit a disconnecting set of size two are those which are ordinary polygons.