Abstract
We consider a special class of bipartite graphs, called chain graphs, defined as \(\{C_3, C_5, 2K_2\}\)-free graphs, that have no repeated Laplacian eigenvalues. Our results include structure theorems, degree constraints and examinations of the corresponding eigenspaces. For example, it occurs that such chain graphs do not contain a triplet of vertices with the same neighbourhood, while those with duplicated vertices (pairs with the same neighbourhood) have additional structural restrictions. As an application, we consider the controllability of multi-agent dynamical systems modelled by graphs under consideration with respect to Laplacian dynamics. We construct particular controllable chain graphs and, in general, provide the minimum number of leading agents as well as their locations in the corresponding graph.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Alazemi A, AnĐelić M, Das KCh, da Fonseca CM (2022) Chain graph sequences and Laplacian spectra of chain graphs, Linear Multilinear Algebra, https://doi.org/10.1080/03081087.2022.2036672
Anđelić M, Brunetti M, Stanić Z (2020) Laplacian controllability for graphs obtained by some standard products. Graphs Combin. 36:1593–1602
Anđelić M, da Fonseca CM, Koledin T, Stanić Z (2013) Sharp spectral inequalities for connected bipartite graphs with maximal \(Q\)-index. Ars Math. Contemp. 6:171–185
Bell FK, Cvetković D, Rowlinson P, Simić SK (2008) Graphs for which the least eigenvalue is minimal, II. Linear Algebra Appl. 429:2168–2179
Brouwer AE, Haemers WH (2012) Spectra of Graphs. Springer, New York
Das KCh, Alazemi A, Anđelić M (2020) On energy and Laplacian energy of chain graphs. Discrete Appl. Math. 284:391–400
Farrugia A, Koledin T, Stanić Z (2022) Controllability of NEPSes of graphs. Linear Multilinear Algebra 70:1928–1941
Hogben L, Reinhart C (2022) Spectra of variants of distance matrices of graphs and digraphs: a survey. La Matematica 1:186–224
Hsu S-P (2016) Controllability of the multi-agent system modelled by the threshold graph with one repeated degree. Systems Control Lett. 97:149–156
Ji Z, Wang Z, Lin H, Wang Z (2009) Interconnection topologies for multi-agent coordination under leader-follower framework. Automatica 45(12):2857–2863
Kailath T (1980) Linear systems. Prentice-Hall, Englewood Cliffs
Kalman RE, Ho YC, Narendra KS (1962) Controllability of linear dynamical systems, Contributions to Differential Equations, Vol. 1
Liu X, Ji Z (2016) Controllability of multiagent systems based on path and cycle graphs. Int. J. Robust Nonlinear Control 28:296–309
Mahadev NVR, Peled UN (1995) Threshold Graphs and Related Topics. North-Holland, New York
Ogata K (2002) Modern control engineering. Prentice-Hall, Upper Saddle River
Rahmani A, Ji M, Mesbahi M, Egerstedt M (2009) Controllability of multi-agent systems from a graph theoretic perspective. SIAM J. Control Opt. 48:162–186
Acknowledgements
This work was supported by Kuwait University, Research Grant No. SM05/20.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Leonardo de Lima.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Alazemi, A., Anđelić, M., Koledin, T. et al. Chain graphs with simple Laplacian eigenvalues and their Laplacian dynamics. Comp. Appl. Math. 42, 6 (2023). https://doi.org/10.1007/s40314-022-02141-5
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s40314-022-02141-5