A generalization of the graph laplacian with application to a distributed consensus algorithm

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In order to describe the interconnection among agents with multi-dimensional states, we generalize the notion of a graph Laplacian by extending the adjacency weights (or weighted interconnection coefficients) from scalars to matrices. More precisely, we use positive definite matrices to denote full multi-dimensional interconnections, while using nonnegative definite matrices to denote partial multi-dimensional interconnections. We prove that the generalized graph Laplacian inherits the spectral properties of the graph Laplacian. As an application, we use the generalized graph Laplacian to establish a distributed consensus algorithm for agents described by multi-dimensional integrators.

Original languageEnglish
Pages (from-to)353-360
Number of pages8
JournalInternational Journal of Applied Mathematics and Computer Science
Issue number2
Publication statusPublished - 2015 Jun 1



  • adjacency weights
  • cooperative control
  • distributed consensus algorithm
  • generalized graph Laplacian
  • graph Laplacian

ASJC Scopus subject areas

  • Applied Mathematics
  • Computer Science (miscellaneous)
  • Engineering (miscellaneous)

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