Research On Related Issues Based On Steiner Distance

The distance of graph is a very important and basic concept in graph theory,which is the basis of studying the distance based graph invariant.Steiner distance is a classic problem in graph theory combinatorial research.The vertex set S is a non-empty subset of connected graph G.Steiner distance of vertex set S is defined as the minimum number of edges of a connected subgraph containing all vertices in S.Steiner k-eccentricity of vertex v in graph G is defined as the maximum value of dG(S)on all S ? V(G)in S,and | S|=k.Among them,the smallest keccentricity is called the Steiner k-radius of the graph G,and the largest k-eccentricity is called the Steiner k-diameter of the graph G.The Steiner Wiener index is a topological index based on distance,which can reflect the structural characteristics and chemical properties of chemical graphs.Therefore,the study of molecular graph topological index has become one of the most active research fields in modern chemical graph theory.Firstly,on the basis of studying the Steiner diameter of corona product and cluster product,the upper and lower bounds of Steiner k-radius of corona product and Steiner k-radius of cluster product are given.And the related theorem is proved according to the definition of Steiner distance of these two product graphs.Secondly,the bounds of Steiner distance and Steiner k-diameter of F-sum graph are obtained,and the Steiner distance of F-sum graph is proved by the definition of Steiner tree.Finally,the Steiner Wiener indices of several chemical graphs are given,and the Steiner kWiener indices of several benzene chain are calculated by using the connectivity of the graphs,where k=3,t-1,t-2.