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Infinite networks and variation of conductance functions in discrete Laplacians

### Abstract

For a given infinite connected graph G = (V, E) and an arbitrary but fixed conductance function c, we study an associated graph Laplacian Δ
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 Δ
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 Δ
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.

© 2015 AIP Publishing LLC

Received 05 June 2014
Accepted 08 April 2015
Published online 21 April 2015

Acknowledgments:
The co-authors thank the following colleagues for helpful and enlightening discussions: Professors Sergii Bezuglyi, Paul Muhly, Myung-Sin Song, Wayne Polyzou, Gestur Olafsson, Keri Kornelson, and members in the Math Physics seminar at the University of Iowa. We are grateful to an anonymous referee for very helpful clarifications and improvements.

Article outline:

I. INTRODUCTION
II. PRELIMINARIES
A. Basic setting
1. Overview of main results in the paper
B. The energy Hilbert spaces ℋ_{
E
}
C. The graph-Laplacian
III. THE KREIN EXTENSION
A. A factorization via *l*
^{2}(*V*′)
B. Further comments
IV. THE SPECTRUM OF THE GRAPH LAPLACIANS
V. VARIATION OF CONDUCTANCE
A. Comparing spectra
B. Comparing harmonic functions, and Dirac masses

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2015-04-21

2016-10-23

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