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Construction of cospectral graphs, signed graphs and \({\mathbb {T}}\)-gain graphs via partial transpose

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Abstract

In the wake of Dutta and Adhikari, who in 2020 used partial transposition in order to get pairs of cospectral graphs, we investigate partial transposition for Hermitian complex matrices. This allows us to construct infinite pairs of complex unit gain graphs (or \({\mathbb {T}}\)-gain graphs) sharing either the spectrum of the adjacency matrix or even the spectrum of all the generalized adjacency matrices. This investigation also sheds new light on the classical case, producing examples that were still missing even for graphs. Partial transposition requires a block structure of the matrix: we interpreted it as if coming from a composition of \({\mathbb {T}}\)-gain digraphs. By proposing a suitable definition of rigidity specifically for \({\mathbb {T}}\)-gain digraphs, we then produce the first examples of pairs of non-isomorphic graphs, signed graphs and \({\mathbb {T}}\)-gain graphs obtained via partial transposition of matrices whose blocks form families of commuting normal matrices. In some cases, the non-isomorphic graphs detected in this way turned out to be hardly distinguishable, since they share the adjacency, the Laplacian and the signless Laplacian spectrum, together with many non-spectral graph invariants.

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Acknowledgements

We thank the anonymous referees for their valuable comments and suggestions, which certainly improved the presentation of the paper. This work has been partially supported by grant ISARP with ref. no. ITAL22051410615 of the National Research Foundation of South Africa. The authors also acknowledge the support received by the National Research Group GNSAGA (Gruppo Nazionale per le Strutture Algebriche, Geometriche e le loro Applicazioni) of Indam.

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Authors and Affiliations

  1. Dipartimento di Matematica e Applicazioni, Università degli Studi di Napoli Federico II, Via Cintia, 80126, Naples, Italy

    Francesco Belardo & Maurizio Brunetti

  2. Dipartimento di Ingegneria, Università degli Studi Niccolò Cusano, Via Don Carlo Gnocchi, 3, 00166, Rome, Italy

    Matteo Cavaleri & Alfredo Donno

Authors
  1. Francesco Belardo
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  2. Maurizio Brunetti
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  3. Matteo Cavaleri
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  4. Alfredo Donno
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Correspondence to Matteo Cavaleri.

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Belardo, F., Brunetti, M., Cavaleri, M. et al. Construction of cospectral graphs, signed graphs and \({\mathbb {T}}\)-gain graphs via partial transpose. J Algebr Comb (2024). https://doi.org/10.1007/s10801-024-01332-z

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  • DOI: https://doi.org/10.1007/s10801-024-01332-z

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