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Frustration index and Cheeger inequalities for discrete and continuous magnetic Laplacians

Lange, Carsten; Liu, Shiping; Peyerimhoff, Norbert; Post, Olaf

Frustration index and Cheeger inequalities for discrete and continuous magnetic Laplacians Thumbnail


Authors

Carsten Lange

Shiping Liu

Olaf Post



Abstract

We discuss a Cheeger constant as a mixture of the frustration index and the expansion rate, and prove the related Cheeger inequalities and higher order Cheeger inequalities for graph Laplacians with cyclic signatures, discrete magnetic Laplacians on finite graphs and magnetic Laplacians on closed Riemannian manifolds. In this process, we develop spectral clustering algorithms for partially oriented graphs and multi-way spectral clustering algorithms via metrics in lens spaces and complex projective spaces. As a byproduct, we give a unified viewpoint of Harary’s structural balance theory of signed graphs and the gauge invariance of magnetic potentials.

Citation

Lange, C., Liu, S., Peyerimhoff, N., & Post, O. (2015). Frustration index and Cheeger inequalities for discrete and continuous magnetic Laplacians. Calculus of Variations and Partial Differential Equations, 54(4), 4165-4196. https://doi.org/10.1007/s00526-015-0935-x

Journal Article Type Article
Acceptance Date Sep 14, 2015
Online Publication Date Nov 5, 2015
Publication Date Dec 1, 2015
Deposit Date Aug 14, 2015
Publicly Available Date Dec 1, 2015
Journal Calculus of Variations and Partial Differential Equations
Print ISSN 0944-2669
Electronic ISSN 1432-0835
Publisher Springer
Peer Reviewed Peer Reviewed
Volume 54
Issue 4
Pages 4165-4196
DOI https://doi.org/10.1007/s00526-015-0935-x

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