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Descriptive Complexity of Graph Spectra

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Logic, Language, Information, and Computation (WoLLIC 2016)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9803))

Abstract

Two graphs are co-spectral if their respective adjacency matrices have the same multi-set of eigenvalues. A graph is said to be determined by its spectrum if all graphs that are co-spectral with it are isomorphic to it. We consider these properties in relation to logical definability. We show that any pair of graphs that are elementarily equivalent with respect to the three-variable counting first-order logic \(C^3\) are co-spectral, and this is not the case with \(C^2\), nor with any number of variables if we exclude counting quantifiers. We also show that the class of graphs that are determined by their spectra is definable in partial fixed-point logic with counting. We relate these properties to other algebraic and combinatorial problems.

We thank Aida Abiad, Chris Godsil, Robin Hirsch and David Roberson for fruitful discussions. This work was supported by CONACyT, EPSRC and The Royal Society.

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Authors and Affiliations

  1. University of Cambridge Computer Laboratory, Cambridge, UK

    Anuj Dawar

  2. Department of Computer Science, University College London, London, UK

    Simone Severini & Octavio Zapata

Authors
  1. Anuj Dawar
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  2. Simone Severini
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  3. Octavio Zapata
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Correspondence to Octavio Zapata .

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Editors and Affiliations

  1. Dept of Mathematics & Statistics, Univ of Helsinki Dept of Mathematics & Statistics, Helsinki, Finland

    Jouko Väänänen

  2. Department of Mathematics and Statistics, University of Helsinki , Helsinki, Finland

    Åsa Hirvonen

  3. Centro de Informática , Recife, Pernambuco, Brazil

    Ruy de Queiroz

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Dawar, A., Severini, S., Zapata, O. (2016). Descriptive Complexity of Graph Spectra. In: Väänänen, J., Hirvonen, Å., de Queiroz, R. (eds) Logic, Language, Information, and Computation. WoLLIC 2016. Lecture Notes in Computer Science(), vol 9803. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-52921-8_12

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  • DOI: https://doi.org/10.1007/978-3-662-52921-8_12

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  • Print ISBN: 978-3-662-52920-1

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