On The Extremal Values Of The Eccentric Distance Sum Of Trees

The eccentric distance sum of graph is an active and important field in graph theo-ry, the research on the eccentric distance sum of tree has become a hot topic in spectral the eccentric distance sum of graph. In 1957, H.Wiener posed the problem of a topo-logical index. This problem has received much attention in mathematics.The wiener index is defined as the sum of all distances between unordered pairs of vertices The eccentricity (v) of a vertex v is the maximum distance from v to any other vertex. DG{v) is the sum of all distances from the vertex v.The eccentric distance sum of G is defined as where sg{v) is the eccentricity of the vertex v and is the maximum distance from v to any other vertex.The specific contents of this paper are as follows:In Section 1, we introduce the development of graph theory, background of re-search, some definitions which are mentioned in this paper, and the summary of this paper; In Section 2, we introduce some lemmas which are mentioned in this paper; In Section 3, we study the maximal eccentric distance sum of n-vertex trees with domina-tion number γ=3.Also, characterize the extremal graph; In Section 4,we determine the trees with the maximal and minimal eccentric distance sums among the n-vertex trees with independence number a and matching number m. Also, the extremal graph of the maximal and the second maximal eccentric distance sums among the n-vertex trees with perfect matching.