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Eigenpolytopes of Distance Regular Graphs

Published online by Cambridge University Press:  20 November 2018

C. D. Godsil
C. D. Godsil*
Affiliation:
Combinatorics and Optimization, University of Waterloo Waterloo, Ontario, N2L 3G1
*
e-mail: [email protected]
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Abstract

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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 $\left| V \right|$ vectors in ${{R}^{m}}$. 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 $X$.

Keywords

05E3005C50
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 50 , Issue 4 , 01 August 1998 , pp. 739 - 755
Copyright
Copyright © Canadian Mathematical Society 1998

Footnotes

Support from a National Sciences and Engineering Council of Canada operating grant is gratefully acknowledged

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