Jordan canonical form of directed Dutch windmill graphs with application to Consensus
Abstract
In this paper, we consider directed Dutch Windmill graphs, a class of graphs composed of multiple cycles sharing a common vertex. For this topology, the spectral analysis of their Laplacian matrix shows that for some eigenvalues, the algebraic multiplicities exceed their geometric multiplicities. Consequently, these Laplacian matrices are not diagonalizable and admit Jordan chains. We derive analytical expressions for the eigenvalues, eigenvectors, and associated Jordan chains. As an application, we address consensus problems in multi-agent systems. In this context, the network topology is represented by a graph 𝒢. The transient responses and final states of the agents are determined by the initial conditions, as well as the eigenvalues and eigenvectors of the associated Laplacian matrix. When considering second-order consensus control, the state matrix derived from the Laplacian may not be diagonalizable due to the graph structure or singular parameter values. We demonstrate that this matrix can be reduced to a Jordan Canonical Form (JCF) and provide explicit analytical expressions for its eigenvalues and generalized eigenvectors.
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PDFDOI: http://dx.doi.org/10.5614/ejgta.2026.14.2.8
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