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BY 4.0 license Open Access Published by De Gruyter Open Access December 2, 2019

A note on the eigenvalue free intervals of some classes of signed threshold graphs

  • Milica Anđelić EMAIL logo , Tamara Koledin and Zoran Stanić
From the journal Special Matrices


We consider a particular class of signed threshold graphs and their eigenvalues. If Ġ is such a threshold graph and Q(Ġ ) is a quotient matrix that arises from the equitable partition of Ġ , then we use a sequence of elementary matrix operations to prove that the matrix Q(Ġ ) – xI (x ∈ ℝ) is row equivalent to a tridiagonal matrix whose determinant is, under certain conditions, of the constant sign. In this way we determine certain intervals in which Ġ has no eigenvalues.

MSC 2010: 05C50; 05C22


[1] C.O. Aguilar, J. Lee, E. Piato, B.J. Schweitzer, Spectral characterizations of anti-regular graphs, Linear Algebra Appl., 557 (2018), 84-104.10.1016/j.laa.2018.07.028Search in Google Scholar

[2] A. Alazemi, M. Anđelić, T. Koledin, Z. K. Stanić, Eigenvalue-free intervals of distance matrices of threshold and chain graphs, Linear Multilinear Algebra, submitted.Search in Google Scholar

[3] A.E. Brouwer, W.H. Haemers, Spectra of graphs, Springer, 2011.10.1007/978-1-4614-1939-6Search in Google Scholar

[4] M. Anđelić, C. M. da Fonseca, Sufficient conditions for positive definiteness of tridiagonal matrices revisited, Positivity, 15 (2011), 155–159.10.1007/s11117-010-0047-ySearch in Google Scholar

[5] D. Cvetković, P. Rowlinson, S.K. Simić, An Introduction to the Theory of Graph Spectra, Cambridge University Press, Cambridge, 2010.10.1017/CBO9780511801518Search in Google Scholar

[6] E. Ghorbani, Eigenvalue-free interval for threshold graphs, Linear Algebra Appl., 583 (2019), 300-305.10.1016/j.laa.2019.08.028Search in Google Scholar

[7] D.P. Jacobs, V. Trevisan, and F. Tura, Eigenvalue location in threshold graphs, Linear Algebra Appl., 439 (2013), 2762-2773.10.1016/j.laa.2013.07.030Search in Google Scholar

[8] D.P. Jacobs, V. Trevisan, F. Tura, Eigenvalues and energy in threshold graphs, Linear Algebra Appl., 465 (2015), 412–425.10.1016/j.laa.2014.09.043Search in Google Scholar

[9] C.R. Johnson, M. Neumann, M.J. Tsatsomeros, Conditions for the positivity of determinants, Linear Multilinear Algebra, 40 (1996), 241–248.10.1080/03081089608818442Search in Google Scholar

[10] V.R. Mahadev, U.N. Peled, Threshold Graphs and Related Topics, North-Holland, Amsterdam, 1995.Search in Google Scholar

[11] Z. Stanić, Inequalities for Graph Eigenvalues, Cambridge University Press, Cambridge, 2015.10.1017/CBO9781316341308Search in Google Scholar

Received: 2019-09-02
Accepted: 2019-11-21
Published Online: 2019-12-02

© 2018 Milica Anđelić et al., published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

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