Study On The Dynamic Characteristics And Isomorphism Discrimination Of Topological Graphs

The isomorphism problem of topological graph is one of the important problems in graph theory.It is of great significance to study the effective isomorphism discrimination algorithm in related fields.In this paper,four equivalent linear dynamic models of topological graph are studied according to the structural characteristics of topological graph.Firstly,mass spring model,which takes the vertex of topological graph as the mass point and the edge as the edge with elastic stiffness coefficient;Secondly,simple pendulum model without inertial coupling,which takes the vertex of topological graph as the simple pendulum and the edge of graph is connected by spring;Thirdly,in the simple pendulum model with inertia coupling,the vertex of the topological graph is regarded as the simple pendulum,the edge of the graph is connected to the simple pendulum with spring,and the suspension point of the simple pendulum is connected with spring;Fourthly,the vertex of the topology graph is regarded as an elastic beam fixed at one end,and the free end of the elastic beam is connected with spring according to the edge of the graph.The equivalent dynamic model of topological graph is independent of the vertex number of topological graph.According to the linear dynamic model and the adjacency matrix of the topological graph,the dynamic equation of the topological graph vibration motion equation is established.The natural frequency(eigenvalue)and mode vector(eigenvector)of the linear dynamic model of the topological graph describe the structural characteristics of the original topological graph.The natural frequency of the dynamic model of the topological graph with mass and stiffness is studied Change of coefficient.According to the topological dynamic model and its characteristics,this paper studies the isomorphic discrimination method of topology,and proposes four isomorphic discrimination algorithms.Firstly,the initial coordinates of two vertices on any edge of the graph are modified,and the free vibration displacement response to the initial coordinates is obtained by using the eigenvalues and eigenvectors of the linear vibration system.Secondly,the harmonic forces of the two vertices are amended,and the forced vibration displacement response is calculated through the dynamic flexibility matrix.Thirdly,the parameters of the elastic beams corresponding to the two vertices or the elastic coefficient of the elastic element corresponding to the edge are modified,and the forced vibration displacement response is obtained by solving the linear system of equations.The mapping relation of the vertexes of any two isomorphic graphs can be set up after the modification of the relevant parameters and obtain of the displacement responses.The dynamic properties of some topological graphs are calculated.Examples show that there are heterogeneous graphs but they have the same natural frequency,which indicates that the dynamic properties are necessary but not sufficient conditions for graph isomorphism,and that there are high re natural frequencies for highly symmetric regular graphs.In this paper,we use the method of isomorphism discrimination to distinguish the isomorphism of some topological graphs.The examples show that there are two cases for heterogeneous topological graphs,the first case is that there are different dynamic characteristics,for which we can directly give the conclusion of isomerism,the second case is that heterogeneous topological graphs have the same dynamic characteristics,but we can not find the vertex mapping relation The results show that it has a good discrimination effect on the isomorphic topological graphs with dispersive natural frequencies,but it may have a misjudgment on the isomorphic topological graphs with high natural frequencies.