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A generalization of the graph Laplacian with application to a distributed consensus algorithm
International Journal of Applied Mathematics and Computer Science, Tome 25 (2015) no. 2, pp. 353-360.

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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.
Keywords: graph Laplacian, generalized graph Laplacian, adjacency weights, distributed consensus algorithm, cooperative control 
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Zhai, G. A generalization of the graph Laplacian with application to a distributed consensus algorithm. International Journal of Applied Mathematics and Computer Science, Tome 25 (2015) no. 2, pp. 353-360. http://geodesic.mathdoc.fr/item/IJAMCS_2015_25_2_a12/

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