Journal article
Infinite networks and variation of conductance functions in discrete Laplacians
Journal of mathematical physics, Vol.56(4), p.43506
04/01/2015
DOI: 10.1063/1.4918646
Abstract
For a given infinite connected graph G = (V, E) and an arbitrary but fixed conductance function c, we study an associated graph Laplacian Delta(c); it is a generalized difference operator where the differences are measured across the edges E in G; and the conductance function c represents the corresponding coefficients. The graph Laplacian (a key tool in the study of infinite networks) acts in an energy Hilbert space H-E computed from c. Using a certain Parseval frame, we study the spectral theoretic properties of graph Laplacians. In fact, for fixed c, there are two versions of the graph Laplacian, one defined naturally in the l(2) space of V and the other in H-E. The first is automatically selfadjoint, but the second involves a Krein extension. We prove that, as sets, the two spectra are the same, aside from the point 0. The point zero may be in the spectrum of the second, but not the first. We further study the fine structure of the respective spectra as the conductance function varies, showing now how the spectrum changes subject to variations in the function c. Specifically, we study an order on the spectra of the family of operators Delta(c), and we compare it to the ordering of pairs of conductance functions. We show how point-wise estimates for two conductance functions translate into spectral comparisons for the two corresponding graph Laplacians, involving a certain similarity: We prove that point-wise ordering of two conductance functions c on E induces a certain similarity of the corresponding (Krein extensions computed from the) two graph Laplacians Delta(c). The spectra are typically continuous, and precise notions of fine-structure of spectrum must be defined in terms of equivalence classes of positive Borel measures (on the real line). Our detailed comparison of spectra is analyzed this way. (C) 2015 AIP Publishing LLC.
Details
- Title: Subtitle
- Infinite networks and variation of conductance functions in discrete Laplacians
- Creators
- Palle Jorgensen - Univ Iowa, Dept Math, Iowa City, IA 52242 USAFeng Tian - Wright State University
- Resource Type
- Journal article
- Publication Details
- Journal of mathematical physics, Vol.56(4), p.43506
- DOI
- 10.1063/1.4918646
- ISSN
- 0022-2488
- eISSN
- 1089-7658
- Publisher
- AMER INST PHYSICS
- Number of pages
- 27
- Language
- English
- Date published
- 04/01/2015
- Academic Unit
- Mathematics
- Record Identifier
- 9984241150402771
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