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Characterization of digraphs with three complementarity eigenvalues

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Abstract

Given a digraph D, the complementarity spectrum of the digraph is defined as the set of complementarity eigenvalues of its adjacency matrix. This complementarity spectrum has been shown to be useful in several fields, particularly in spectral graph theory. The differences between the properties of the complementarity spectrum for (undirected) graphs and for digraphs make the study of the latter of particular interest, and characterizing strongly connected digraphs with a small number of complementarity eigenvalues is a nontrivial problem. Recently, strongly connected digraphs with one and two complementarity eigenvalues have been completely characterized. In this paper, we study strongly connected digraphs with exactly three elements in the complementarity spectrum, ending with a complete characterization. This leads to a structural characterization of general digraphs having three complementarity eigenvalues.

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Notes

  1. Also referred as complementary eigenvalues.

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Acknowledgements

This research is part of the doctoral studies of F. Cubría. V. Trevisan acknowledges partial support of CNPq Grants 409746/2016-9 and 310827/2020-5, and FAPERGS Grant PqG 17/2551-0001. M. Fiori and D. Bravo acknowledge the financial support provided by ANII, Uruguay. F. Cubría thanks the doctoral scholarship from CAP-UdelaR. We thank Lucía Riera for the help with the figures.

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Authors and Affiliations

  1. R &D Department, Tryolabs, Montevideo, Uruguay

    Diego Bravo

  2. Instituto de Matemática y Estadística “Rafael Laguardia”, Facultad de Ingeniería, Universidad de la Republica, Montevideo, Uruguay

    Florencia Cubría & Marcelo Fiori

  3. Instituto de Matemática, Universidade Federal do Rio Grande do Sul, Porto Alegre, Brazil

    Vilmar Trevisan

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  1. Diego Bravo
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  2. Florencia Cubría
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  3. Marcelo Fiori
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  4. Vilmar Trevisan
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Correspondence to Florencia Cubría.

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Bravo, D., Cubría, F., Fiori, M. et al. Characterization of digraphs with three complementarity eigenvalues. J Algebr Comb 57, 1173–1193 (2023). https://doi.org/10.1007/s10801-023-01218-6

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