On the definiteness of graph Laplacians with negative weights: Geometrical and passivity-based approaches

Yongxin Chen, Sei Zhen Khong, Tryphon T. Georgiou

Research output: Chapter in Book/Report/Conference proceedingConference contribution

29 Scopus citations


The positive semidefiniteness of Laplacian matrices corresponding to graphs with negative edge weights is studied. Two alternative proofs to a result by Zelazo and Bürger (Theorem 3.2), which provides upper bounds on the magnitudes of the negative weights in terms of effective resistances within which to ensure definiteness of the Laplacians, are provided. Both proofs are direct and intuitive. The first employs purely geometrical arguments while the second relies on passivity arguments and the laws of physics for electrical circuits. The latter is then used to establish consensus in multi-agent systems with generalized high-order dynamics. A numerical example is given at the end of the paper to highlight the result.

Original languageEnglish (US)
Title of host publication2016 American Control Conference, ACC 2016
PublisherInstitute of Electrical and Electronics Engineers Inc.
Number of pages6
ISBN (Electronic)9781467386821
StatePublished - Jul 28 2016
Event2016 American Control Conference, ACC 2016 - Boston, United States
Duration: Jul 6 2016Jul 8 2016

Publication series

NameProceedings of the American Control Conference
ISSN (Print)0743-1619


Other2016 American Control Conference, ACC 2016
Country/TerritoryUnited States

Bibliographical note

Funding Information:
This research was supported in part by the NSF under Grant ECCS-1509387, the AFOSR under Grant FA9550-12-1-0319, and the Institute for Mathematics and its Applications with funds provided by the NSF.

Publisher Copyright:
© 2016 American Automatic Control Council (AACC).


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