The Resistance Distance Of Several Composite Graphs

The resistance distance R(u,v) of the vertexandin the connected graphi G s defined as the effective resistance of these two points in the corresponding electrical network N after a resistance of 1 ohm is added to each edge.The resistance diameter D_r(G)of graphis defined as the maximum resistance distance between all pairs of vertices in the graph.The Kirchhoff index Kf(g)of graphis defined as the sum of the resistance distances between all pairs of vertex in graph.In this paper,we mainly studied the resistance distance and its correlation exponent of linear chain of complete graphs,and the resistance diameter of grid and its derivative graphs.The distribution is as follows:In the first chapter,we briefly introduced the relevant background,current research status at home and abroad,and main research content.In the second chapter,some necessary concepts and lemmas related to graph theory and electrical networks were given.In the third chapter,we mainly studied the resistance distance and its correlation exponent of the linear chain of complete graphs.By introducing the concept of nega-tive resistance,combining with substitution principle and star network transformation,a simple electrical network equivalent to the complete graph linear chain was construct-ed.The resistance distance between any pairs of vertex in a complete graph linear chain was obtained by eliminating the principle of series-parallel star-triangle transformation.Secondly,the Kirchhoff index of the special complete graph linear chain were given.In the fourth chapter,we mainly studied the resistance diameters of grid and its derivative graphs.With the help of conservation of energy and the minimum energy consumption in the electrical network,combined with the inequality properties of ordered graphs,the resistance distance of some pairs of vertex in the lexicographic product of ordered graphs and arbitrary graphs was studied.The resistance diameters of grid and their derivative graphs were given by using a similar method when the ordered graph and arbitrary graph were restricted to the path graph.In the fifth chapter,we summarized the main results of this paper and raised the relevant unsolved problems.