Abstract
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
x-\frac{|S|}{v}\one
is an eigenvector for
A(X)
with eigenvalue
τ
. 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.