Some Results Of Graphs With An Involution

This thesis consists of four parts.In the Chapter 1,we first give some basic concepts and notations which are needed in this thesis,then simply introduce the background of the topic.In the Chapter 2,we express the Kirchhoff index and resistance distances of a graph with an involution in terms of that of two smaller graphs (?) and (?).As applications of the results obtained in Chapter 2,the resistance distances and Kirchhoff indices of several kinds of graphs are computed,such as double graphs,almost-complete graphs and almostcomplete bipartite graphs.In the last Chapter of this thesis,we compute the resistance distances between any two vertices in a linear polyomino chain and linear hexagonal chain,respectively.At the same time,the asymptotic properties of the resistance distances in these two kinds of graphs are discussed.