Jordan canonical form of directed Dutch windmill graphs with application to Consensus

Francois Bateman (Centre de Recherche de l'Ecole de l'air - Ecole de l'air et de l'espace - France)
Pierre Bagnara (Centre de Recherche de l'Ecole de l'Air, Ecole de l'air et de l'espace, Salon de Provence, France)
Fabrice Barache (Ecole de l'air et de l'espace, Salon de Provence, France)

Electronic Journal of Graph Theory and Applications (EJGTA)
Vol 14, No 2 (2026): Electronic Journal of Graph Theory and Applications

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 ?. 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.