Frustration index and Cheeger inequalities for discrete and continuous magnetic Laplacians

Carsten Lange; Shiping Liu; Norbert Peyerimhoff; Olaf Post

doi:10.1007/s00526-015-0935-x

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arXiv · 1502.06299

Frustration index and Cheeger inequalities for discrete and continuous magnetic Laplacians

Carsten Lange·Shiping Liu·Norbert Peyerimhoff·Olaf Post

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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.

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Carsten Lange, Shiping Liu, Norbert Peyerimhoff, Olaf Post. 2015-02-23. Frustration index and Cheeger inequalities for discrete and continuous magnetic Laplacians. https://doi.org/10.1007/s00526-015-0935-x

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