Abstract view
Eigenpolytopes of distance regular graphs


Published:19980801
Printed: Aug 1998
Abstract
Let $X$ be a graph with vertex set $V$ and let $A$ be
its adjacency matrix. If $E$ is the matrix representing orthogonal
projection onto an eigenspace of $A$ with dimension $m$, then $E$ is
positive semidefinite. Hence it is the Gram matrix of a set of $V$
vectors in $\re^m$. We call the convex hull of a such a set of vectors
an eigenpolytope of $X$. The connection between the properties of this
polytope and the graph is strongest when $X$ is distance regular and,
in this case, it is most natural to consider the eigenpolytope
associated to the second largest eigenvalue of $A$. The main result
of this paper is the characterisation of those distance regular graphs
$X$ for which the $1$skeleton of this eigenpolytope is isomorphic to
$X$.