How do the eigenvalues of the Laplacian matrix affect route to synchronization patterns?

In cluster synchronization, network nodes are divided into synchronized groups before the whole network gets synchronized. This phenomenon is crucial in understanding the mechanism behind the synchronization of realworld and man-made complex networks like neuronal networks and power grids. The Laplacian matrix and its eigenvalues provide helpful information about the networks' synchronization, robustness, and controllability. Here, analyzing the relation between the Laplacian matrix eigenvalues and cluster synchronization demonstrates that the intensity of the eigenvalues has significant importance on the clusters' appearance. Results show that considering groups of equal eigenvalues yields the appearing of clusters in the network. So, this technique allows the ability to design networks with the desired number of clusters with defined cluster size. Synchronization of the clusters results in a plateau in the order parameter evolution. Furthermore, studying two different chaotic systems shows that this relationship depends on the systems' dynamics. Here, the eigenvector centrality tool is utilized to examine the existence of clusters besides the graph representation.