Topological Properties Of Corona Graph And Their Applications In Complex Networks

Complex systems are ubiquitous in the real world.They include biological systems,social systems,and climate and environment,and the rest.The theory and application of complex systems are of great significance in many research areas.Complex networks,as the abstract topology of complex systems,can visually reflect the interrelationship among system objects.Graph is an effective tool to study the topological structure of complex networks,and the properties of graph invariants play an important role in describing the topological properties of complex networks.In this paper,both the ABC index and Wiener index,two kinds of Laplacian spectra of the corona graphs,were first studied,and then the ABC energy of specific graphs was calculated.Accordingly,the application of graph theory in community detection of complex networks was analyzed,which provided a new perspective for community detection and community structure analysis using the graph invariants.The main work is as follows:(1)The ABC index of the corona of two graphs was studied,it was proved that the ABC index of the corona graph decreases after the edge deletion operation of the two graphs.And the upper bound of ABC index of the edge corona of two graphs was obtained.In this paper,the ABC index of the corona of two trees was discussed.Combining the structure characteristics of the graph and the property of inequality,the upper bound of the ABC index of the corona of two trees was obtained and the extremal graph was described.It provided great insights for solving the quantitative analysis problems of other types of corona graphs.(2)The ABC index of the edge corona of two graphs was researched.It was proved that the ABC index of the edge corona graph decreases after the edge deletion operation of the two graphs.And the upper bound of ABC index of the edge corona of two graphs was obtained.(3)The Wiener index of the corona of three graphs was considered by analyzing the structure of the corona graph and classifying the vertex pairs of different types in the graph.It was concluded that the Wiener index of(G1(?)G2)(?)G3 was related not to the structure of graphs G1 and G2,but to the number of vertices and edges of the three graphs,and it was monotonically decreasing with the number of edges of graphs G1 and G2.(4)Taking the weighted corona graph—a topological model closer to the real network—as the research object,the signless Laplacian spectrum and the normalized Laplacian spectrum of two types of weighted corona graphs were studied.It was proved that the spectra depicted were all the eigenvalues corresponding to the weighted corona graphs,and all the eigenvectors corresponding to them were given.The conclusion of normalized Laplacian spectrum had been applied and popularized,aiming to provide theoretical references for widely applying the spectrum theory into practice.(5)The concept of graph energy was introduced,and then a new graph energy(ABC energy)was considered.By using the particularity of ABC matrix,the ABC energy of several special graphs,including friendship graph and windmill graph,were calculated.The ABC energy after edge removal of special graphs was also deduced.(6)The application of spectra of graphs in detecting the community of complex networks was analyzed and summarized.Then two clustering algorithms closely related to the spectra of graphs were introduced,and lastly new similarity metrics on the basis of the existing algorithms were proposed.It offered theoretical insights in applying other graph invariants to community detection.