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The theta number of simplicial complexes

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Abstract

We introduce a generalization of the celebrated Lovász theta number of a graph to simplicial complexes of arbitrary dimension. Our generalization takes advantage of real simplicial cohomology theory, in particular combinatorial Laplacians, and provides a semidefinite programming upper bound of the independence number of a simplicial complex. We consider properties of the graph theta number such as the relationship to Hoffman’s ratio bound and to the chromatic number and study how they extend to higher dimensions. Like in the case of graphs, the higher dimensional theta number can be extended to a hierarchy of semidefinite programming upper bounds reaching the independence number. We analyze the value of the theta number and of the hierarchy for dense random simplicial complexes.

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Authors and Affiliations

  1. Institut de Mathématiques de Bordeaux, UMR 5251, Université de Bordeaux, 351 Cours de la Libération, 33400, Talence, France

    Christine Bachoc & Alberto Passuello

  2. Mathematisches Institut, Universität zu Köln, Weyertal 86-90, 50931, Köln, Germany

    Anna Gundert

Authors
  1. Christine Bachoc
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  2. Anna Gundert
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  3. Alberto Passuello
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Correspondence to Anna Gundert.

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Bachoc, C., Gundert, A. & Passuello, A. The theta number of simplicial complexes. Isr. J. Math. 232, 443–481 (2019). https://doi.org/10.1007/s11856-019-1880-8

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  • DOI: https://doi.org/10.1007/s11856-019-1880-8

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