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Spectral preorder and perturbations of discrete weighted graphs

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To Hagen Neidhardt in memoriam.

Abstract

In this article, we introduce a geometric and a spectral preorder relation on the class of weighted graphs with a magnetic potential. The first preorder is expressed through the existence of a graph homomorphism respecting the magnetic potential and fulfilling certain inequalities for the weights. The second preorder refers to the spectrum of the associated Laplacian of the magnetic weighted graph. These relations give a quantitative control of the effect of elementary and composite perturbations of the graph (deleting edges, contracting vertices, etc.) on the spectrum of the corresponding Laplacians, generalising interlacing of eigenvalues. We give several applications of the preorders: we show how to classify graphs according to these preorders and we prove the stability of certain eigenvalues in graphs with a maximal d-clique. Moreover, we show the monotonicity of the eigenvalues when passing to spanning subgraphs and the monotonicity of magnetic Cheeger constants with respect to the geometric preorder. Finally, we prove a refined procedure to detect spectral gaps in the spectrum of an infinite covering graph.

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Notes

  1. Note that in this article we switched to the more standard (and for our purposes more convenient) notation that an edge e (also called an arrow) always has its oppositely oriented counterpart \({\bar{e}}\) in E as in [15, 38]; in our older articles (e.g. in [17]) we used the convention that E contains only one arrow (not its opposite arrow), hence in our older notation E together with \(\partial e=(\partial _+e,\partial _-e)\) determines already an orientation of the graph.

  2. Note that in some references the standard weight is also called normalised.

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Acknowledgements

We thank a referee of this article for her/his very thorough review, comments and suggestions.

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Authors and Affiliations

  1. Department of Mathematics, University Carlos III de Madrid, Avda. de la Universidad 30, 28911, Leganés, Madrid, Spain

    John Stewart Fabila-Carrasco & Fernando Lledó

  2. Instituto de Ciencias Matemáticas (CSIC-UAM-UC3M-UCM), Madrid, Spain

    John Stewart Fabila-Carrasco & Fernando Lledó

  3. Fachbereich 4-Mathematik, Universität Trier, 54286, Trier, Germany

    Olaf Post

Authors
  1. John Stewart Fabila-Carrasco
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  2. Fernando Lledó
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  3. Olaf Post
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Correspondence to Fernando Lledó.

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Communicated by Y. Giga.

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JSFC and FLl were supported by Spanish Ministry of Economy and Competitiveness through project DGI MTM2017-84098-P, from the Severo Ochoa Programme for Centres of Excellence in R& D (SEV-2015-0554) and from the Spanish National Research Council, through the Ayuda extraordinaria a Centros de Excelencia Severo Ochoa (20205CEX001).

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Fabila-Carrasco, J.S., Lledó, F. & Post, O. Spectral preorder and perturbations of discrete weighted graphs. Math. Ann. 382, 1775–1823 (2022). https://doi.org/10.1007/s00208-020-02091-5

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  • DOI: https://doi.org/10.1007/s00208-020-02091-5

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