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Curvature and higher order Buser inequalities for the graph connection Laplacian

Liu, Shiping; Muench, Florentin; Peyerimhoff, Norbert

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Authors

Shiping Liu

Florentin Muench



Abstract

We study the eigenvalues of the connection Laplacian on a graph with an orthogonal group or unitary group signature. We establish higher order Buser type inequalities, i.e., we provide upper bounds for eigenvalues in terms of Cheeger constants in the case of nonnegative Ricci curvature. In this process, we discuss the concepts of Cheeger type constants and a discrete Ricci curvature for connection Laplacians and study their properties systematically. The Cheeger constants are defined as mixtures of the expansion rate of the underlying graph and the frustration index of the signature. The discrete curvature, which can be computed efficiently via solving semidefinite programming problems, has a characterization by the heat semigroup for functions combined with a heat semigroup for vector fields on the graph.

Citation

Liu, S., Muench, F., & Peyerimhoff, N. (2019). Curvature and higher order Buser inequalities for the graph connection Laplacian. SIAM Journal on Discrete Mathematics, 33(1), 257-305. https://doi.org/10.1137/16m1056353

Journal Article Type Article
Acceptance Date Dec 17, 2019
Online Publication Date Feb 5, 2019
Publication Date Feb 28, 2019
Deposit Date Jan 22, 2019
Publicly Available Date Apr 11, 2019
Journal SIAM Journal on Discrete Mathematics
Print ISSN 0895-4801
Electronic ISSN 1095-7146
Publisher Society for Industrial and Applied Mathematics
Peer Reviewed Peer Reviewed
Volume 33
Issue 1
Pages 257-305
DOI https://doi.org/10.1137/16m1056353

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Copyright Statement
© 2019 Society for Industrial and Applied Mathematics




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