The spectra of lifted digraphs

Abstract

We present a method to derive the complete spectrum of the lift \(\varGamma ^\alpha \) of a base digraph \(\varGamma \), with voltage assignment \(\alpha \) on a (finite) group G. The method is based on assigning to \(\varGamma \) a quotient-like matrix whose entries are elements of the group algebra \(\mathbb {C}[G]\), which fully represents \(\varGamma ^{\alpha }\). This allows us to derive the eigenvectors and eigenvalues of the lift in terms of those of the base digraph and the irreducible characters of G. Thus, our main theorem generalizes some previous results of Lovász and Babai concerning the spectra of Cayley digraphs.

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Correspondence to M. A. Fiol.

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The first author has also received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie Grant Agreement No. 734922.

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Research of the first two authors is supported by AGAUR under Project 2017SGR1087. The third author acknowledges support from the research Grants APVV 0136/12, APVV-15-0220, VEGA 1/0026/16, and VEGA 1/0142/17.

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Dalfó, C., Fiol, M.A. & Širáň, J. The spectra of lifted digraphs. J Algebr Comb 50, 419–426 (2019). https://doi.org/10.1007/s10801-018-0862-y

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Keywords

  • Digraph
  • Adjacency matrix
  • Regular partition
  • Quotient digraph
  • Spectrum
  • Lifted digraph

Mathematics Subject Classification

  • 05C20
  • 05C50
  • 15A18