The consensus of complex networks has attracted the attention of many scholars. The graph operation is a common method to construct complex networks, which is helpful in studying the consensus of complex networks. Based on the corona networks G₁◦G₂, this study gives different weights to the edges of G₁◦G₂ to obtain the weighted corona networks G̃1◦G̃2 and studies the consensus of G̃1◦G̃2. The consensus of the networks can be measured by coherence. First, the Laplacian polynomial of G̃1◦G̃2 is derived by using the properties of an orthogonal matrix. Second, the relationship between the first-order coherence of G̃1◦G̃2 and G₁ is deduced by using the relevant properties of the determinant and the conclusion of polynomial coefficients and the principal minors of the matrix. Third, the join operation is introduced to further simplify the analytical formula of network coherence. Finally, a specific network example is used to verify the validity of the conclusion.

In this paper, several multi-layer-coupled star-composed networks with similar symmetrical structures are defined by using the theory of graph operation. The supra-Laplacian matrix of the corresponding multi-layer networks is obtained according to the master stability equation (MSF). Two important indexes that reflect the synchronizability of these kinds of networks are derived in the case of bounded and unbounded synchronized regions. The relationships among the synchronizability, the number of layers, the length of the paths, the branchings, and the interlayer and intralayer coupling strengths in the two cases are studied. At the same time, the simulation experiments are carried out with the MATLAB software, and the simulated images of the two symmetrical structure networks’ synchronizability are compared. Finally, the factors affecting the synchronizability of multi-layer-coupled star-composed networks are found. On this basis, optimization schemes are given to improve the synchronizability of multi-layer-coupled star-composed networks and the influences of the number of central nodes on the networks’ synchronizability are further studied.

The topology structure of multi-layer networks is highly correlated with the robustness of consensus. This paper investigates the influence of different interlayer edge connection patterns on the consensus of the two-layer ring networks. Two types of two-layer ring network models are first considered: one is a kind of two-layer ring network with two linked edges between layers (Networks-a), and the other is a kind of two-layer ring network with three linked edges between layers (Networks-b). Using the Laplacian spectrum, the consensus of the network model is derived. The simulation experiments are used to demonstrate the influence of different interlayer edge connection patterns on the consensus of networks. To determine the best edge connection pattern for Networks-a and Networks-b, the number of nodes in a single-layer ring network is denoted by n. The best edge connection pattern for Networks-a is 1 & $\left[\right(n+2)/2]$. Furthermore, n is subdivided into $3k,3k+1,3k+2$, and the best edge connection patterns of Networks-b are near 1 & $k+1$ & $2k+1$.