Some results on the complete sigraphs with exactly three non-negative eigenvalues

Document Type : Original paper

Authors

Department of Mathematics, Karaj Branch, Islamic Azad University, Karaj, Iran

Abstract

Let $(K_{n},H^-)$ be a complete sigraph of order $n$ whose negative edges induce a subgraph $H$. In this paper, we characterize $(K_n,H^-)$ with exactly 3 non-negative eigenvalues, where $H$ is a non-spanning two-cyclic subgraph of $K_n$.

Keywords

Main Subjects


[1] S. Akbari, S. Dalvandi, F. Heydari, and M. Maghasedi, On the multiplicity of −1 and 1 in signed complete graphs, Util. Math. 116 (2020), 21–32.
[2] S. Akbari, S. Dalvandi, F. Heydari, and M. Maghasedi, On the eigenvalues of signed complete graphs, Linear Multilinear Algebra 67 (2019), no. 3, 433–441.
https://doi.org/10.1080/03081087.2017.1403548
[3] F. Belardo, M. Brunetti, M. Cavaleri, and A. Donno, Constructing cospectral signed graphs, Linear Multilinear Algebra 69 (2021), no. 14, 2717–2732.
https://doi.org/10.1080/03081087.2019.1694483
[4] A.E. Brouwer and W.H. Haemers, Spectra of Graphs, Springer Science & Business Media, 2011.
[5] M. Brunetti and A. Ciampella, Signed bicyclic graphs with minimal index, Commun. Comb. Optim. 8 (2023), no. 1, 207–241.
https://doi.org/10.22049/cco.2022.27346.1241
[6] M. Brunetti and Z. Stanić, Ordering signed graphs with large index, Ars Math. Contemp. 22 (2022), no. 4, 595–608.
https://doi.org/10.26493/1855-3974.2714.9b3
[7] M. Brunetti and Z. Stanić, Unbalanced signed graphs with extremal spectral radius or index, Comput. Appl. Math. 41 (2022), no. 3, Article number: 118.
https://doi.org/10.1007/s40314-022-01814-5
[8] D. Cvetković, P. Rowlinson, and S. Simić, An Introduction to the Theory of Graph Spectra, Cambridge University Press, London, 2009.
[9] S. Dalvandi, F. Heydari, and M. Maghasedi, Signed complete graphs with exactly m non-negative eigenvalues, Bull. Malays. Math. Sci. Soc. 45 (2022), no. 5, 2107–2122.
https://doi.org/10.1007/s40840-022-01331-y
[10] S. Dalvandi, F. Heydari, and M. Maghasedi, A characterization of (kn, u−) in the class l(3), Ric. Mat. (2024), in press
https://doi.org/10.1007/s11587-023-00844-3
[11] M.R. Oboudi, Characterization of graphs with exactly two non-negative eigenvalues, Ars Math. Contemp. 12 (2016), no. 2, 271–286.
[12] M. Petrović, Graphs with a small number of nonnegative eigenvalues, Graphs Combin. 15 (1999), no. 2, 221–232.
https://doi.org/10.1007/s003730050042
[13] J.H. Smith, Symmetry and multiple eigenvalues of graphs, Glas. Mat. Ser. III 12 (1977), no. 1, 3–8.
[14] M. Souri, F. Heydari, and M. Maghasedi, Maximizing the largest eigenvalues of signed unicyclic graphs, Discrete Math. Algorithms Appl. 12 (2020), no. 2, Article ID: 2050016.
https://doi.org/10.1142/S1793830920500160
[15] Z. Stanić, Some relations between the skew spectrum of an oriented graph and the spectrum of certain closely associated signed graphs, Rev. de la Union Mat. Argentina 63 (2022), no. 1, 41–50.
https://doi.org/10.33044/revuma.1914
[16] A. Torgašev, Graphs with exactly two negative eigenvalues, Math. Nachr. 122 (1985), no. 1, 135–140.
https://doi.org/10.1002/mana.19851220113
[17] T. Zaslavsky, Matrices in the theory of signed simple graphs, Advances in Discrete Mathematics and Applications, Ramanujan Mathematical Society Lecture Notes Series 13, Ramanujan Mathematical Society, Mysore, 2010, pp. 207–229.