Research On Reciprocal Distance Laplacian Spectrum Of Graphs And Its Application In Complex Networks

In recent years,with the rapid development of science and technology,human society has entered the era of the internet.Today,the world is filled with various complex networks,and people’s lives and production are inseparable from these diverse types of networks.The internet age has made the connections between people closer,and closer connections mean that local influences are more likely to spread.From another perspective,this indicates that negative events such as internet paralysis and epidemics will be more destructive.Therefore,the increasingly networked human society requires us to have a deeper understanding of the behavior and characteristics of various complex networks.With the continuous deepening of research on complex networks,more and more scholars are paying attention to the significant impact of a small number of important nodes on the overall network function.Quickly and effectively mining important nodes in complex networks has become a hot research topic today.The scale and structure of the network are constantly changing,so it is of great practical significance to mine the important nodes in the network quickly and effectively.The study of complex networks cannot be separated from the relevant theories in graph theory.Graph theory has been widely applied in various complex networks such as transportation networks,socio-economic networks,communication networks,logistics networks,sensor networks,the internet,and other fields.A concrete network can be abstracted as a graph composed of a node set and an edge set: each point in the node set can be represented as an abstract city,a logistics center,a transportation hub,and a person;Each edge in the edge set can represent the distance between two cities or transportation hubs,the relationship between two logistics centers,the relationship between two people,etc.This can transform network problems into graphs for research.The topological index can reflect the local characteristics of complex networks,which can reflect the structure property of the graph effectively.In complex networks,the topological index is closely related to the overall characteristics of the complex network,such as damage resistance and robustness.By calculating the topological index of nodes in the network and reordering their importance according to the value,the reliability of complex networks can be evaluated effectively and the important nodes can be protected.The failure of important nodes will cause serious damage to the whole actual network,so the study of topological index can improve the overall characteristics of complex networks.These features can be used to rank the importance of nodes,so as to find the important nodes in the complex network,and then protect the complex network.In graph theory,the topological structure and algebraic properties of graphs are studied mainly through the eigenvalues,eigenvectors and chromatic numbers of various matrices(adjacency matrix,distance matrix,reciprocal distance matrix,reciprocal distance(signless Laplacian matrix,etc.)of graphs.In this paper,the reciprocal distance matrix,the reciprocal distance(signless)Laplacian matrix of graphs are used to study their topological properties,and on this basis,the properties of complex network topology are studied.The main conclusions of this paper are as follows:(1)In this paper,some properties of the reciprocal distance Laplacian spectrum are demonstrated,and the reciprocal distance Laplacian characteristic polynomials for several classes of graphs is derived.Finally,the upper bound of the reciprocal distance Laplacian eigenvalues of the second to last and third to last under a given chromatic number obtained,and the extreme value diagram is depicted.(2)Some bounds of the reciprocal distance signless Laplacian matrix spectra in terms of various graph parameters are obtained.The relationship between the reciprocal distance signless Laplacian matrix and the reciprocal distance matrix of graph G is also established,and the lower bound of the reciprocal distance signless Laplacian matrix of the bipartite graph is obtained.(3)This article introduces the generalized reciprocal distance matrix from the reciprocal distance Laplace matrix and the reciprocal distance signless Laplace matrix.By using the reciprocal distance degree and the second reciprocal distance degree,the upper and lower bounds on the spectral radius of the generalized reciprocal distance matrix are given.And the bounds of the spectral radius of the generalized reciprocal distance matrix of the line graph are given.(4)This paper introduces several classical methods to study the centrality of nodes,and proposes to use reciprocal distance energy,reciprocal distance Laplacian energy and reciprocal distance signless Laplacian energy to study the centrality of nodes.A communication network node importance evaluation method based on the correlation characteristics between nodes is proposed to address the potential changes in network topology caused by node removal in existing node importance evaluation methods.The maximum degree matrix variation method defined by this method considers the impact of the connection relationships between different nodes in the network on the importance of nodes and measures the importance of deleted nodes by using the change in their corresponding maximum degree matrix after deletion.Research has found that this method is superior to traditional degree centrality in determining node importance.