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Combinatorics, Probability and Computing Combinatorics, Probability and Computing

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Eigenvalue Ratios of Non-Negatively Curved Graphs

Part of: Graph theory Global differential geometry

Published online by Cambridge University Press:  23 May 2018

SHIPING LIU  and
NORBERT PEYERIMHOFF
SHIPING LIU
Affiliation:
Wu Wen-Tsun Key Laboratory of Mathematics, Chinese Academy of Sciences, School of Mathematical Sciences, University of Science and Technology of China, Hefei 230026, Anhui Province, China (e-mail: spliu@ustc.edu.cn)
NORBERT PEYERIMHOFF
Affiliation:
Department of Mathematical Sciences, Durham University, DH1 3LE Durham, UK (e-mail: norbert.peyerimhoff@durham.ac.uk)
Corresponding
E-mail address:
spliu@ustc.edu.cn
norbert.peyerimhoff@durham.ac.uk
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Abstract

We derive an optimal eigenvalue ratio estimate for finite weighted graphs satisfying the curvature-dimension inequality CD(0, ∞). This estimate is independent of the size of the graph and provides a general method to obtain higher-order spectral estimates. The operation of taking Cartesian products is shown to be an efficient way for constructing new weighted graphs satisfying CD(0, ∞). We also discuss a higher-order Cheeger constant-ratio estimate and related topics about expanders.

MSC classification

Primary: 05C50: Graphs and linear algebra (matrices, eigenvalues, etc.)
Secondary: 53C23: Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 27 , Issue 5 , September 2018 , pp. 829 - 850
Copyright
Copyright © Cambridge University Press 2018 

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