Research On Inverses Of Graphs And Resistance Distances Of Graphs

Let be a connected graph.The inverse of is defined as the graph whose adjacency matrix is the inverse of the adjacency matrix of .The resistance distance between any two vertices of is defined as the net effective resistance between them if each edge of is replaced by a unite resistor.The Kirchhoff index of is defined as the sum of resistance distances between all pairs of vertices.In this thesis,we mainly study the inverse,resistance distance and the Kirchhoff index of graphs.The whole thesis is divided into five chapters,as given in the following.First,in the first chapter,we introduce definitions and notations that are used in the thesis,and survey the research progress on the inverse,resistance distance and the Kirchhoff index of graphs.In chapter 2,we characterize the inverse of a class of non-bipartite graphs with unique perfect matching.For a non-bipartite graph with a unique perfect matching,if there exist an edge such that the deletion of the edge results a bipartite graph with a unique perfect matching,then we could obtain exact expression for the inverse of its adjacency matrix,so that the inverse of such kind of graph is characterized.As applications,we characterize the inverse of the graph which is obtained from the odd cycle of order 9)by adding pendent edges to its 9)-2 consecutive vertices.Then,in chapter 3,we calculate resistance distances in linear hexagonal chains.By methods and techniques in electrical network theory as well as local sum rules on resistance distances,exact expression for resistance distances in linear hexagonal chains is derived.For a triangulation graph which is embedded in a orientable surface,the graph obtained by first adding a new vertex to each face of and then connecting each of the newly added vertex to the vertices of the corresponding face is called the vertex-face graph of .In the forth chapter,by the relation of resistance distances between the vertex-face graph and the original graph,we obtain exact expression for the Kirchhoff index.It turns out that the Kirchhoff index of the vertex-face graph can be expressed in terms of parameters of ,such as the number of vertices,the number of faces,and the Kirchhoff index of .Finally,making use of comparison theorem on the Kirchhoff index of ,-isomers,we characterized extremal phenylene chains with respect to the Kirchhoff index.It is shown that the linear phenylene chain is the unique graph with maximal Kirchhoff index,while the minimal Kirchhoff index is obtained only when the phenylene chain is an “all-kink” chain.