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Curvature and higher order Buser inequalities for the graph connection Laplacian.

Liu, Shiping and Muench, Florentin and Peyerimhoff, Norbert (2019) 'Curvature and higher order Buser inequalities for the graph connection Laplacian.', SIAM journal on discrete mathematics., 33 (1). pp. 257-305.

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.

Item Type:Article
Full text:Publisher-imposed embargo
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Status:Peer-reviewed
Publisher Web site:https://doi.org/10.1137/16M1056353
Publisher statement:© 2019 Society for Industrial and Applied Mathematics
Date accepted:17 December 2019
Date deposited:23 January 2019
Date of first online publication:05 February 2019
Date first made open access:No date available

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