Eigenvalues Of Several Kinds Of Joint Graphs And Weighted Treelike Networks And Their Applications

In this dissertation,according to the concept of joint graphs,we mainly study network consistency of a class of weighted star-composed networks with weight factor,find the adjacency spectrum of the two types of joint graphs,and also obtain the expression of characteristic polynomials of adjacency and Laplacian matrix of the weighted treelike networks.In Chapter 1,we introduce the development background of complex networks and joint graphs,also introduce the definitions of the corona graph and Kronecker product of matrices.Then we review the concepts of network coherence of weighted networks,these are the foundation for computing we study later.In Chapter 2,we study a class of weighted star-composed networks.Firstly,we introduce the generation of star-composed networks and determine the weighting method through the idea of corona graph.Secondly,we use the Kronecker product between the matrices to express the Laplacian matrix,and then obtain the relationship of Laplacian eigenvalues at two successive generations.Finally,we analyze the first-order and second-order network coherence quantifying as the sum and square sum of reciprocals of all nonzero Laplacian eigenvalues.We get that the consistency scale of the weighted star-composed networks changes with the weight,and it has no relation with the number of nodes in the initial graph.In Chapter 3,we study the eigenvalues of adjacency matrices?referred to as adjacency eigenvalues?for Quasi-corona SG-vertex join and multiple SG-vertex join graphs.Knowing the adjacent eigenvalues of₁G andG₂,when₁G is a regular graph,we obtain the expression of the characteristic polynomials of the adjacent matrix of G₁?G ₂andG₁?G₂.When both₁G and₂G are regular graphs,we obtain all adjacent eigenvalues of two kinds of joint graphs.In Chapter 4,firstly,a class of weighted treelike networks with a weight factor are introduced.Then the relationships of adjacency or the Laplacian matrix at two successive generations are obtained.Finally,according to the operation of the block matrix,we obtain the analytic expressions of the characteristic polynomial of the adjacency or the Laplacian matrix of weighted treelike networks.In Chapter 5,we summarize the dissertation according to the research contents.