Independent sets in association schemes

Let $X$ be $k$-regular graph on $v$ vertices and let $\tau$ denote the least eigenvalue of its adjacency matrix $A(X)$. If $\alpha (X)$ denotes the maximum size of an independent set in $X$, we have the following well known bound: $$\alpha (X)\le \frac{v}{1-\frac{k}{\tau }}\mathrm{.}$$ 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 $$x-\frac{\mathrm{\mid }S\mathrm{\mid }}{v}\text{one}$$ is an eigenvector for $A(X)$ with eigenvalue $\tau$. In this paper we show how this can be used to characterise the maximal independent sets in certain classes of graphs. As a corollary we show that a graph defined on the partitions of $\{1,\mathrm{.}\mathrm{.}\mathrm{.},9\}$ with three cells of size three is a core.