Which digraphs with ring structure are essentially cyclic?
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Cites
Harary, F. (1969) · Cited by 10,764 · see citing works
Consensus and Cooperation in Networked Multi-Agent Systems
Olfati-Saber, R., Fax, J. A., & Murray, R. M. (2007) · Proceedings of the IEEE · doi:10.1109/jproc.2006.887293 · Cited by 10,541 · cited 1× · see citing works
“In particular, in the decentralized control of multi-agent systems [10, 19, 21, 23], the absence of non-real Laplacian eigenvalues of the communication digraph implies that the simplest consensus algorithms are devoid of oscillations.”
BackgroundDistributed Consensus in Multi-vehicle Cooperative Control
Ren, W., & Beard, R. W. (2008) · Communications and Control Engineering · doi:10.1007/978-1-84800-015-5 · Cited by 2,826 · see citing works
cacute, C., Vs, M., Doob, M., & Sachs, H. (1980) · Cited by 1,478 · see citing works
A Survey of Matrix Theory and Matrix Inequalities
H., A. S., Marcus, M. D., & Minc, H. (1965) · Mathematics of Computation · doi:10.2307/2004116 · Cited by 1,448 · see citing works
A Survey of Matrix Theory and Matrix Inequalities.
Bellman, R., Marcus, M. D., & Minc, H. (1965) · American Mathematical Monthly · doi:10.2307/2315999 · Cited by 843 · see citing works
Laplacians and the Cheeger Inequality for Directed Graphs
Chung, F. (2005) · Annals of Combinatorics · doi:10.1007/s00026-005-0237-z · Cited by 535 · see citing works
Eigenvalues of the Laplacian of a graph ∗
Anderson, W. N., & Morley, T. D. (1985) · Linear and Multilinear Algebra · doi:10.1080/03081088508817681 · Cited by 486 · see citing works
On the spectra of nonsymmetric Laplacian matrices
Agaev, R., & Chebotarev, P. (2005) · Linear Algebra and its Applications · doi:10.1016/j.laa.2004.09.003 · Cited by 177 · see citing works
Self-citationTHE MATRIX-FOREST THEOREM AND MEASURING RELATIONS IN SMALL SOCIAL GROUPS
Chebotarev, P., & Shamis, E. (1997) · Automation and Remote Control · Cited by 169 · see citing works
Self-citationAlgebraic connectivity of directed graphs
Wu, C. W. (2005) · Linear and Multilinear Algebra · doi:10.1080/03081080500054810 · Cited by 161 · see citing works
Forest matrices around the Laplacian matrixPreprint
Chebotarev, P., & Agaev, R. (2012) · Cited by 107 · see citing works
Self-citationOn Proximity Measures for Graph VerticesPreprint
Chebotarev, P., & Shamis, E. (2006) · arXiv (Cornell University) · doi:10.48550/arxiv.math/0602073 · Cited by 97 · see citing works
Self-citationSpanning tree formulas and chebyshev polynomials
Boesch, F. T., & Prodinger, H. (1986) · Graphs and Combinatorics · doi:10.1007/bf01788093 · Cited by 92 · see citing works
The Matrix of Maximum Out Forests of a Digraph and Its ApplicationsPreprint
Agaev, R., & Chebotarev, P. (2006) · arXiv (Cornell University) · doi:10.48550/arxiv.math/0602059 · Cited by 91 · see citing works
Self-citationKernels of Directed Graph Laplacians
Caughman, J. S., & Veerman, J. J. P. (2006) · The Electronic Journal of Combinatorics · doi:10.37236/1065 · Cited by 90 · see citing works
Coordination in multiagent systems and Laplacian spectra of digraphs
Chebotarev, P. Y., & Agaev, R. P. (2009) · Automation and Remote Control · doi:10.1134/s0005117909030126 · Cited by 90 · see citing works
Self-citationForest matrices around the Laplacian matrix
Chebotarev, P., & Agaev, R. (2002) · Linear Algebra and its Applications · doi:10.1016/s0024-3795(02)00388-9 · Cited by 88 · see citing works
Self-citationSellami, T., Berriri, H., Mimouni, M. F., Jelassi, S., & Darcherif, A. (2014) · HAL · Cited by 73 · see citing works
Chebyshev polynomials and spanning tree formulas for circulant and related graphs
Zhang, Y., Yong, X., & Golin, M. J. (2005) · Discrete Mathematics · doi:10.1016/j.disc.2004.10.025 · Cited by 71 · see citing works
Veerman, J. J. P., Lafferriere, G., Caughman, J. S., & Williams, A. (2005) · Journal of Statistical Physics · doi:10.1007/s10955-005-6999-9 · Cited by 61 · see citing works
Spanning Forests of a Digraph and Their Applications
Agaev, R. P., & Chebotarev, P. Y. (2001) · Automation and Remote Control · doi:10.1023/a:1002862312617 · Cited by 57 · see citing works
Self-citationBounds for eigenvalues of doubly stochastic matrices
Fiedler, M. (1972) · Linear Algebra and its Applications · doi:10.1016/0024-3795(72)90011-0 · Cited by 44 · see citing works
On $W$-bases of directed graphs
Fiedler, M., & Sedláček, J. (1958) · Časopis pro pěstování matematiky · doi:10.21136/cpm.1958.108301 · Cited by 30 · see citing works
O $W$-basích orientovaných grafů
Fiedler, M., & Sedláček, J. (1958) · Časopis pro pěstování matematiky · Cited by 9 · see citing works
Flagged first, then by citations
3 of the 28 references couldn’t be matched to a known paper.
Research works printed but not linked: 1
- P. Chebotarev, Laplacian matrices and essentially cyclic weighted digraphs, in: Abstracts of the 13th Conference of the International Linear Algebra Society, Free University of Amsterdam, 2006, pp. 60-61.
Cited by
Sorted by citations
Comments on "Consensus and cooperation in networked multi-agent systems"
Chebotarev, P. (2010) · Proceedings of the IEEE · doi:10.1109/jproc.2010.2049911 · Cited by 18 · see citing works
Self-citationScale fragilities in localized consensus dynamics
Tegling, E., Bamieh, B. A., & Sandberg, H. (2023) · Automatica · doi:10.1016/j.automatica.2023.111046 · Cited by 18 · see citing works
Tight bound on parameter of surplus-based averaging algorithm over balanced digraphs
Kawamura, S., Cai, K., Ye, M., & Lin, Z. (2018) · International Journal of Control · doi:10.1080/00207179.2018.1535200 · Cited by 9 · see citing works
Scalable second-order consensus of hierarchical groups
Wang, J., Liu, J., Xiao, F., Xi, N., Mo, L., & Zheng, Y. (2025) · Systems & Control Letters · doi:10.1016/j.sysconle.2025.106032 · Cited by 3 · see citing works
On singularity and properties of eigenvectors of complex Laplacian matrix of multidigraphs
Barik, S., Reddy, S. U., & Sahoo, G. (2023) · AKCE International Journal of Graphs and Combinatorics · doi:10.1080/09728600.2023.2234014 · Cited by 3 · see citing works
Complex adjacency spectra of digraphs
Sahoo, G. (2019) · Linear and Multilinear Algebra · doi:10.1080/03081087.2019.1591337 · Cited by 3 · see citing works
Second-Order Agents on Ring Digraphs
Parsegov, S., & Chebotarev, P. (2018) · International Conference on System Theory, Control and Computing · doi:10.1109/icstcc.2018.8540656 · Cited by 4 · see citing works
Self-citationHierarchical Cyclic Pursuit: Algebraic Curves Containing the Laplacian Spectra
Parsegov, S. E., Chebotarev, P., Shcherbakov, P., & Ibáñez, F. M. (2023) · IEEE Transactions on Control of Network Systems · doi:10.1109/tcns.2023.3237495 · Cited by 3 · see citing works
Self-citationStability and Controllability of Double Integrator Consensus Systems in Heterogeneous Networks
Goldin, D. (2013) · DepositOnce · doi:10.14279/depositonce-3768 · Cited by 3 · see citing works
The Laplacian spectrum of some digraphs obtained from the wheel
Li, H., Su, L., & Liu-rong, Z. (2012) · Discussiones Mathematicae Graph Theory · doi:10.7151/dmgt.1612 · Cited by 2 · see citing works
Further results on digraphs with completely real Laplacian spectra
Li, H.-H., & Su, L. (2015) · Applied Mathematics and Computation · doi:10.1016/j.amc.2015.05.102
- Authors
- Agaev, R., & Chebotarev, P.
- Affiliations
- Russian Academy of Sciences (RU)
- Corresponding author
- Chebotarev, Pavel
- ORCID
- Chebotarev, Pavel
- Venue
- Advances in Applied Mathematics
- Publisher
- Elsevier BV
- Published
- 2010-08
- Volume
- 45
- Issue
- 2
- Pages
- 232-251
- ISSN
- 0196-8858 (ISSN-L)
- OpenAlex
- W1991753382
- Cited by
- 11
- References
- 28
- Citations by year
- 2025: 1 · 2023: 3 · 2019: 1 · 2018: 2 · 2015: 1 · 2013: 1 · 2012: 1
- FWCI
- 1.19
- Type
- journal article
- Language
- en
- Open access
- bronze
- doi.org
- Licence
- Locations
- Advances in Applied Mathematics (publishedVersion)
- arXiv (Cornell University) (submittedVersion)
- arXiv (Cornell University)
- Funders
- Russian Foundation for Basic Research (09-07-00371)
- Topics
- Physical Sciences › Mathematics › Geometry and Topology › Graph theory and applications
- Physical Sciences › Computer Science › Computational Theory and Mathematics › Matrix Theory and Algorithms
- Physical Sciences › Computer Science › Computer Vision and Pattern Recognition › Graph Theory and Algorithms
- Keywords
- Laplacian spectrum, non-real eigenvalues, Chebyshev polynomials of the second kind, polynomial roots, weighted directed graphs, spanning trees, decentralized control, communication networks
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