Journal of Operator Theory
Volume 72, Issue 1, Summer 2014 pp. 71-86.
Spectral comparisons between networks with different
conductance functions
Authors:
Palle E.T. Jorgensen (1) and Erin P.J. Pearse
Author institution: (1) University of Iowa, Iowa City, IA 52246-1419,
U.S.A.
(2) California Polytechnic University, San Luis Obispo, CA 93407-0403,
U.S.A.
Summary: For an infinite network consisting of a graph with
edge weights prescribed
by a given conductance function $c$, we consider the effects of replacing
these weights with a new function $b$ that satisfies $b \leqslant c$ on
each edge. In particular, we compare the corresponding energy spaces
and the
spectra of the Laplace operators acting on these spaces. We use
these
results to derive estimates for effective resistance on the two
networks,
and to compute a spectral invariant for the canonical embedding
of one
energy space into the other.
DOI: http://dx.doi.org/10.7900/jot.2012oct05.1978
Keywords: Dirichlet form, graph energy, unbounded discrete
Laplacian, weighted graph, spectral graph theory, effective resistance,
harmonic analysis, Hilbert space, reproducing kernels
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