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1. CJM 2000 (vol 52 pp. 1057)
The Spectrum of an Infinite Graph In this paper, we consider the (essential) spectrum of the discrete
Laplacian of an infinite graph. We introduce a new quantity for an
infinite graph, in terms of which we give new lower bound estimates of
the (essential) spectrum and give also upper bound estimates when the
infinite graph is bipartite. We give sharp estimates of the
(essential) spectrum for several examples of infinite graphs.
Keywords:infinite graph, discrete Laplacian, spectrum, essential spectrum Categories:05C50, 58G25 |
2. CJM 1998 (vol 50 pp. 739)
Eigenpolytopes of distance regular graphs 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
$X$.
Categories:05E30, 05C50 |