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Eigenpolytopes of distance regular graphs

 Printed: Aug 1998
  • C. D. Godsil
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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 semi-definite. 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 $X$.
MSC Classifications: 05E30, 05C50 show english descriptions Association schemes, strongly regular graphs
Graphs and linear algebra (matrices, eigenvalues, etc.)
05E30 - Association schemes, strongly regular graphs
05C50 - Graphs and linear algebra (matrices, eigenvalues, etc.)

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