A notion of generalized graph Laplacian and its application to distributed consensus algorithm

In order to describe the interconnection among agents with multi-dimensional states, we generalize the notion of graph Laplacian by extending the adjacency weights from positive scalars to positive definite matrices. We prove that the generalized graph Laplacian inherits the spectral properties of the graph Laplacian. As an application example, we use the generalized graph Laplacian to establish a distributed consensus algorithm for agents described by multi-dimensional integrators.