Research On Several Kinds Of Graph Parameters Based On Resistance Distance

The resistance distance R_G(x,y)between two vertices x and y of a graph G is defined as the effective resistance of the two vertices in the electrical network constructed from the graph G by replacing each edge with a unit resistor.The sum of resistance distances between all pairs of vertices in graph G is called the Kirchhoff index.The maximum of the resistance distances between all pairs of vertices in graph G is called the resistance diameter.The resistance spectrum RS(G)of a graph G is defined as the multiset of the resistance distances of all pairs of vertices in the graph.This thesis mainly studied the resistance distance and(degree sum,degree product)Kirchhoff index of the power graph of generalized quaternion group,the relationship between the resistance diameters of the cactus graph and its line graph,and the determination of the resistance spectrum of the distance regular graph.The content of this thesis is organized as follows:In Chapter 1,the research background and significance of this thesis,the research progress at home and abroad,the preliminary basic knowledge and the main work of this thesis were introduced.In Chapter 2,the power graph of the generalized quaternion group was studied,and the resistance distance between any pairs of vertices in the graph and the(degree sum,degree product)Kirchhoff index of the graph were obtained.In Chapter 3,it was proved that the resistance diameter of the line graph of a cactus graph is not more than the resistance diameter of the cactus graph,and it was experimentally verified that a similar relationship holds for any non-isomorphic simple graph G with no more than 11 vertices.In Chapter 4,the problem of determining the resistance spectrum of distance regular graphs was investigated,and it was shown that the complete graphs and cycles are determined by the resistance spectrum.Two distance regular graphs have the same resistance spectrum if and only if they have the same intersection array.As an application,a set of graphs of size 32548 was found,where all graphs in the set share a common resistance spectrum.In addition,three methods for constructing infinitely many pairs of graphs with the same resistance spectrum were found.In Chapter 5,we have summarized the main results of this thesis and propose further research direc-tions.