  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: \[
\alpha(X) \le\frac{v}{1-\frac{k}{\tau}}. \] 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 \[ x-\frac{|S|}{v}\one \] 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
$\{1,...,9\}$ with three cells of size three is a core.
