The Laplacian Spectrum, Kirchhoff Index, and the Number of Spanning Trees of the Linear Heptagonal Networks

Let H n be the linear heptagonal networks with 2 n heptagons. We study the structure properties and the eigenvalues of the linear heptagonal networks. According to the Laplacian polynomial of H n , we utilize the method of decompositions. Thus, the Laplacian spectrum of H n is created by eigenvalues of a pair of matrices: L A and L S of order numbers 5 n + 1 and 4 n + 1 n ! / r ! n − r ! , respectively. On the basis of the roots and coefficients of their characteristic polynomials of L A and L S , we get not only the explicit forms of Kirchhoff index but also the corresponding total number of spanning trees of H n .

Synchronizability of Multi-Layer Dual-Center Coupled Star Networks

In the research on complex networks, synchronizability is a significant measurement of network nature. Several research studies center around the synchronizability of single-layer complex networks and few studies on the synchronizability of multi-layer networks. Firstly, this paper calculates the Laplacian spectrum of multi-layer dual-center coupled star networks and multi-layer dual-center coupled star–ring networks according to the master stability function (MSF) and obtains important indicators reflecting the synchronizability of the above two network structures. Secondly, it discusses the relationships among synchronizability and various parameters, and numerical simulations are given to illustrate the effectiveness of the theoretical results. Finally, it is found that the two sorts of networks studied in this paper are of the same synchronizability, and compared with that of a single-center network structure, the synchronizability of two dual-center structures is relatively weaker.

Robustness of network coherence in asymmetric unicyclic graphs

In this paper, we propose a family of unicyclic graphs to study robustness of network coherence quantified by the Laplacian spectrum, which measures the extent of consensus under the noise. We adjust the network parameters to change the structural asymmetries with an aim of studying their effects on the coherence. Using the graph’s structures and matrix theories, we obtain closed-form solutions of the network coherence regarding network parameters and network size. We further show that the coherence of the asymmetric graph is higher than the corresponding symmetric graph and also compare the consensus behaviors for the graphs with different asymmetric structures. It displays that the coherence of the unicyclic graph with one hub is better than the graph with two hubs. Finally, we investigate the effect of degree of hub nodes on the coherence and find that bigger difference of degrees leads to better coherence.

The Normalized Laplacians, Degree-Kirchhoff Index, and the Complexity of Möbius Graph of Linear Octagonal-Quadrilateral Networks

The normalized Laplacian plays an indispensable role in exploring the structural properties of irregular graphs. Let L n 8,4 represent a linear octagonal-quadrilateral network. Then, by identifying the opposite lateral edges of L n 8,4 , we get the corresponding Möbius graph M Q n 8,4 . In this paper, starting from the decomposition theorem of polynomials, we infer that the normalized Laplacian spectrum of M Q n 8,4 can be determined by the eigenvalues of two symmetric quasi-triangular matrices ℒ A and ℒ S of order 4 n . Next, owing to the relationship between the two matrix roots and the coefficients mentioned above, we derive the explicit expressions of the degree-Kirchhoff indices and the complexity of M Q n 8,4 .