Research On Eccentric Distance Sum Of Trees And Unicyclic Graphs

In 2002, eccentric distance sum(EDS) was proposed as a new molecular topological index which is defined as()=∑()()∈where () is the eccentricity of the vertex and () = ∑ ∈(, ) is the sum of all distances from the vertex .Through the experiment methods, Gupta e.t.al[3] proved that the eccentric distance sum has higher discriminating power than the other topological indices with respect to predicting both biological activity and physical property. After this, S.Sardana, A.K. Madan[57] confirmed that the eccentric distance sum has more accurate in calculation the antioxidant activity of nitroxides. So it is significant to carry out the research on EDS in-depth.Based on the previous studies and basic structural characteristics of graphs, this paper mainly discusses three aspects in the way of graphic transformation: the trees having the maximal EDS among n-vertex trees with maximum degree ?; the trees having the maximal EDS among n-vertex trees with domination number 4; the running of leaves of corresponding vertices in unicyclic graph. The research methods are including Classified discussion, Exclusive method and Reduction to absurdity. The results are as follows:In Chapter 1, it mainly introduces the background and the existing results of the research.In Chapter 2, it il ustrates the basic definitions, symbols and the related lemmas mentioned in this paper.In Chapter 3, the trees having the maximal EDS among n-vertex trees with maximum degree ? are characterized. The extremal trees is a special spider graph which satisfied 2= ??= 1. On this basis, the upper bound of the EDS among n-vertex trees with maximum degree ? is given.In Chapter 4, the trees having the maximal EDS among n-vertex trees with domination number 4 are discussed. Based on the maximum degree values of non-domination vertex, these trees are divided into 3 types. Then the EDS of 10.?;102?, ?;102?/ is proved to be maximal among n-vertex trees with domination number 4 according to the discussion of each type.In Chapter 5: the running of leaves of corresponding vertices in unicyclic graph is discussed.