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BY 4.0 license Open Access Published by De Gruyter Open Access December 23, 2019

Inertias of Laplacian matrices of weighted signed graphs

  • K. Hassani Monfared , G. MacGillivray EMAIL logo , D. D. Olesky and P. van den Driessche
From the journal Special Matrices


We study the sets of inertias achieved by Laplacian matrices of weighted signed graphs. First we characterize signed graphs with a unique Laplacian inertia. Then we show that there is a sufficiently small perturbation of the nonzero weights on the edges of any connected weighted signed graph so that all eigenvalues of its Laplacian matrix are simple. Next, we give upper bounds on the number of possible Laplacian inertias for signed graphs with a fixed flexibility τ (a combinatorial parameter of signed graphs), and show that these bounds are sharp for an infinite family of signed graphs. Finally, we provide upper bounds for the number of possible Laplacian inertias of signed graphs in terms of the number of vertices.

MSC 2010: 05C50; 05C22; 15B35


[1] A. Arenas, A. Díaz-Guilera, J. Kurths, and Y. Moreno. Synchronization in complex networks. Physics Reports, 469:93–153, 2008.10.1016/j.physrep.2008.09.002Search in Google Scholar

[2] W. Chen, D. Wang, J. Liu, T. Başar, and L. Qiu. On spectral properties of signed Laplacians for undirected graphs. 2017 IEEE 56th Annual Conference on Decision and Control (CDC), pages 1999–2002, 2017.10.1109/CDC.2017.8263941Search in Google Scholar

[3] L. Pan, H. Shao, and M. Mesbahi. Laplacian dynamics on signed networks. 2016 IEEE 55th Conference on Decision and Control (CDC), pages 891–896, 2016.10.1109/CDC.2016.7798380Search in Google Scholar

[4] F. Belardo and M. Brunetti. Connected signed graphs L-cospectral to signed 1-graphs. Linear and Multilinear Algebra, 67:2410–2426, 2019.10.1080/03081087.2018.1494122Search in Google Scholar

[5] X. Wang, D. Wong, and F. Tian. Signed graphs with cut points whose positive inertia indexes are two. Linear Algebra and its Applications, 539:14–27, 2018.10.1016/j.laa.2017.09.014Search in Google Scholar

[6] J. Bronski and L. DeVille. Spectral theory for dynamics on graphs containing attractive and repulsive interactions. SIAM Journal of Applied Mathematics, 74(1):83–105, 2014.10.1137/130913973Search in Google Scholar

[7] C. Poignard, T. Pereira, and J. P. Pade. Spectra of Laplacian matrices of weighted graphs: structural genericity properties. SIAM Journal of Applied Mathematics, 78:372–394, 2018.10.1137/17M1124474Search in Google Scholar

[8] J. Bronski, L. DeVille, and P. Koutsaki. The spectral index of signed Laplacians and their structural stability. arXiv:1503.01069 [math.DS].Search in Google Scholar

[9] W. Chen, D. Wang, J. Liu, T. Başar, K. H. Johansson, and L. Qiu. On semidefiniteness of signed laplacians with application to microgrids. IFAC-PapersOnLine, 49:97–102, 2016.10.1016/j.ifacol.2016.10.379Search in Google Scholar

[10] J.A. Bondy and U.S.R. Murty. Graph Theory with Applications. North Holland, New York, NY, 1976.10.1007/978-1-349-03521-2Search in Google Scholar

[11] R. Tarjan. Depth-first search and linear graph algorithms. SIAM Journal on Computing, 1:146–160, 1972.10.1137/0201010Search in Google Scholar

[12] G. Shi, C. Altafini, and J.S. Baras. Dynamics over signed networks. SIAM Review, 61(2):229–257, 2019.10.1137/17M1134172Search in Google Scholar

Received: 2019-06-21
Accepted: 2019-12-06
Published Online: 2019-12-23

© 2019 K. Hassani Monfared et al., published by De Gruyter

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

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