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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.

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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