Study On Spectral Properties Of Resistance Matrix Of Graphs

In 1993,Klein and Randi? studied the application of resistance distance in chemical graph theory from the perspective of distance function.A unit resistance is placed on each edge of a connected graph to form a resistance network.The effective resistance between vertices in the resistance network is called the resistance distance of the graph.The resistance matrix of connected graph is a matrix composed of resistance distance,and the maximum eigenvalue of resistance matrix is the spectral radius of resistance matrix.The resistance distance of graph is not only an important research topic in chemical graph theory,but also developed into the fields of complex network,graph structure analysis,graph spanning tree counting,graph theory and graph algorithm in recent years.The first chapter of this paper mainly introduces the research background and research status of resistance distance,resistance matrix and spectral radius of resistance matrix.The second chapter mainly introduces the basic knowledge,symbol description and some classical results related to graph theory,matrix theory and resistance distance,which lays the foundation for the main results of this paper.In Chapter 3,firstly,the influence of a kind of edge grafting transformation on the spectral radius of the resistance matrix of a unicyclic graph is studied.Secondly,according to the influence of the edge grafting transformation on the spectral radius of the resistance matrix,several extreme graph classes of the spectral radius of the resistance matrix of a unicyclic graph are given.Finally,the expression of the characteristic polynomial of the resistance matrix of a unicyclic graph with arbitrary girth is given,The expression is helpful to simplify the solution of the spectral radius of the resistance matrix.In Chapter 4,the bounds of the spectral radius of the resistance matrix are given by using the maximum degree of the graph and some resistance centrality indexes.