Bravo, Diego; Cubría, Florencia; Fiori, Marcelo; Trevisan, Vilmar Complementarity spectrum of digraphs. (English) Zbl 1468.05145 Linear Algebra Appl. 627, 24-40 (2021). Summary: In this paper we study the complementarity spectrum of digraphs, with special attention to the problem of digraph characterization through this complementarity spectrum. That is, whether two non-isomorphic digraphs with the same number of vertices can have the same complementarity eigenvalues. The complementarity eigenvalues of matrices, also called Pareto eigenvalues, has led to the study of the complementarity spectrum of (undirected) graphs and, in particular, the characterization of undirected graphs through these eigenvalues is an open problem. We characterize the digraphs with one and two complementarity eigenvalues, and we give examples of non-isomorphic digraphs with the same complementarity spectrum. Cited in 1 Document MSC: 05C50 Graphs and linear algebra (matrices, eigenvalues, etc.) 05C20 Directed graphs (digraphs), tournaments Keywords:complementarity spectrum; complementary spectrum; digraph characterization PDFBibTeX XMLCite \textit{D. Bravo} et al., Linear Algebra Appl. 627, 24--40 (2021; Zbl 1468.05145) Full Text: DOI arXiv References: [1] Adly, S.; Rammal, H., A new method for solving second-order cone eigenvalue complementarity problems, J. Optim. Theory Appl., 165, 563-585 (2015) · Zbl 1321.90137 [2] Brualdi, R. A., Spectra of digraphs, Linear Algebra Appl., 432, 2181-2213 (2010) · Zbl 1221.05177 [3] Pinto da Costa, A.; Martins, J.; Figueiredo, I.; Júdice, J., The directional instability problem in systems with frictional contacts, Comput. Methods Appl. Mech. Eng., 193, 357-384 (2004) · Zbl 1075.74596 [4] Pinto da Costa, A.; Seeger, A., Cone-constrained eigenvalue problems: theory and algorithms, Comput. Optim. Appl., 45, 25-57 (2010) · Zbl 1193.65039 [5] Cvetković, D.; Doob, M.; Sachs, H., Spectra of Graphs: Theory and Applications (1998), Wiley: Wiley New York [6] van Dam, E. R.; Haemers, W. H., Which graphs are determined by their spectrum?, Linear Algebra Appl., 373, 241-272 (2003) · Zbl 1026.05079 [7] Facchinei, F.; Pang, J. S., Finite-Dimensional Variational Inequalities and Complementarity Problems (2007), Springer Science & Business Media [8] Fernandes, R.; Judice, J.; Trevisan, V., Complementary eigenvalues of graphs, Linear Algebra Appl., 527, 216-231 (2017) · Zbl 1365.05171 [9] Harary, F.; King, C.; Mowshowitz, A.; Read, R. C., Cospectral graphs and digraphs, Bull. Lond. Math. Soc., 321-328 (1971) · Zbl 0224.05125 [10] Lin, H.; Shu, J., A note on the spectral characterization of strongly connected bicyclic digraphs, Linear Algebra Appl., 436, 2524-2530 (2012) · Zbl 1238.05163 [11] Pinheiro, L. K.; Souza, B. S.; Trevisan, V., Determining graphs by the complementary spectrum, Discuss. Math., Graph Theory, 40, 607-620 (2020) · Zbl 1433.05205 [12] Sachs, H., Beziehungen zwischen den in einem Graphen enthaltenen Kreisen und seinem charakteristischen Polynom, Publ. Math. (Debr.), 11, 119-134 (1964) · Zbl 0137.18103 [13] Seeger, A., Eigenvalue analysis of equilibrium processes defined by linear complementarity conditions, Linear Algebra Appl., 292, 1-14 (1999) · Zbl 1016.90067 [14] Seeger, A., Complementarity eigenvalue analysis of connected graphs, Linear Algebra Appl., 543, 205-225 (2018) · Zbl 1387.05160 [15] Von Collatz, L.; Sinogowitz, U., Spektren endlicher Grafen, Abh. Math. Semin. Univ. Hamb., 21, 63-77 (1957) · Zbl 0077.36704 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.