Research On The Related Invariants Of Weighted Adjacency Matrix Of Graphs And Its Application In Complex Network

As one of the main research directions in graph theory and combinatorial matrix theory,spectral graph theory focuses on depicting the structure of graphs using the spectra of matrices,and studying the intrinsic connections between the various graph invariants and the spectra of graphs.Among the studies on graph invariants in spectral graph theory,spectral radius,spectral energy and various topological indices are important research contents,that attract the extensive attention of scholars at home and abroad,and their research has not only promoted and enriched graph theory,but also has very broad applications in physics,chemistry,biology,and complex networks.Especially in complex networks,node centrality algorithms,random walk algorithms,community mining algorithms and propagation influence maximization algorithms based on graph invariants provide more effective mathematical tools and methods for further study of complex network topological properties.The matrix studied in this thesis is the weighted adjacency matrix.Firstly,the upper and lower bounds of the weighted spectral radius and the weighted energy are calculated,and the Nordhaus-Gaddum-type relation of the weighted spectral radius is derived.Secondly,the weighted spectrum of two combinatorial graphs and the weighted Estrada index of some special graphs are calculated,in addition,the upper and lower bounds of the weighted Estrada index of general graphs are calculated.Finally,the weighted adjacency matrix is applied to complex networks and combined with the idea of multiattribute decision-making to identify the critical nodes in the network.The main research contents of this thesis are as follows:(1)The upper and lower bounds of the spectral radius and energy of the weighted adjacency matrix are calculated based on the maximum degree,the minimum degree,the number of vertices,the number of edges and various topological indices,and the extremal graphs are described when the upper and lower bounds are reached,on this basis,the Nordhaus-Gaddum-type relation of weighted spectral radius is obtained.(2)The weighted spectrum of the subdivision-vertex join graph and subdivisionedge join graph of regular graphs are calculated.The weighted Estrada index is defined and the weighted Estrada index of complete bipartite graphs,the windmill graphs,the friendship graphs and the -splitting graphs of regular graphs are calculated.In addition,some upper and lower bounds of the weighted Estrada index was calculated.(3)The weighted adjacency matrix is applied to complex networks,and the Entropy Weight Improving-Technique for Order Preference by Similarity to an Ideal Solution method is proposed to identify critical nodes.In this method,local centrality is proposed based on the influence of nodes themselves and the contribution of neighboring nodes,the neighborhood weighted degree centrality is defined based on the weighting degree of the neighboring nodes,the global centrality is defined based on the edge weight,the KS of the neighboring nodes and the shortest effective distance.The three centrality methods are taken as attribute sets of nodes,and after assigning weights to the three attributes using the entropy weight method,the critical nodes in the network are identified by the virtual worst solution improved Technique for Order Preference by Similarity to Ideal Solution method.In order to verify the ability of the proposed method to identify critical nodes,it is compared with eight centrality methods in eleven real network data sets.The experimental results show that the proposed method has better performance and applicability than the comparison method in terms of discriminative ability,accuracy and accuracy of high ranking nodes.