On The Strong Diameter And Radius Of Some Strong Product Graphs

In an undirected connected graph G or a directed connected graph D,the minimum number of edges that a vertex u passes through to another vertex v is the distance from u to v.In view of the asymmetry of distances in directed graphs,Chartrand et al.proposed the concept of strong distance between two vertices and v in a directed graph-the strong distance sd(u,v)between u and v is the minimum number of strongly connected subgraph arcs containing u and v.Where the maximum value of the distance from u to all vertices is the strong eccentricity se(v),the minimum value of the strong eccentricity among all vertices is the strong radius srad(D),and the maximum value is the strong diameter sdiam(D).Orienting all edges of a bipartite connected simple undirected graph G such that there exists a directed path between any two vertices,the resulting directed graph D is a strongly directed graph of G.Therefore,four strong distance parameters of the undirected graph can be obtained:the minimum strong radius srad(G),the maximum strong radius SRAD(G),the minimum strong diameter sdiam(G),and the maximum strong diameter SDIAM(G).In this paper,we focus on the strong radius and the strong diameter for the strong product graphs,and the specific research content includes two parts:(1)Upper and lower bounds on the minimum and maximum strong radius and strong diameter of the general strong product are given using the radius,diameter and order of the factor graph and by combining the basic properties of the strong product.(2)Select the line graphs of complete graphs,paths,and star graphs as factor graphs,and give the upper and lower bounds or exact values of the minimum and maximum strong radius and strong diameters of the strong products of these special graphs.In the process of constructing directed graph,a strong product directed network model based on path is given.The topology of the strong product graph is closely related to its factor graph,which can preserve the good properties of the factor graph and has great prospect for use in network design.Therefore,the study of strong radius and strong diameter of the strong product graph can better measure the topology of directed networks and provide better graph parameters for network design,which has good theoretical significance and application prospects.