The Sum Of All Distances And The Average Distance In A Connected Graph

The sum of distances in graph is just the sum of distances between all pairs of vertices of graph. This concept, introduced by the chemist Wiener, is a quite successful tool for designing quantitative structure-property relations in organic chemistry. As early as in 1947 in American chemist Harold Wiener proposed the modeling of certain physico-chemical properties of hydrocarbon molecules by means of the sum of distances. From then, the research in this field has never been terminated.In chemistry the sum of distances is one of most frequently-used concepts in constructing molecular model by theory of graph. It is closed related to another quantity-mean distance, denoted the average value of the distances between all pairs of vertices in graph, it is a measure of the "compactness" of a graph, it had already been initially used in architecture as a tool for the evaluation of floor plans. Further more, it is used as a measure of the efficiency of a communications network.In a network model, the time used in transmitting information from one vertex to another vertex and the decreasing degree of the signal is always proportional to the length of the line the information must cover. Diameter reflects the worst possible condition while the mean distance reflects its average condition.In a word this two quantities as an important parameter of graph have been paid attention and further research by many experts. Based on the earlier times research on this two quantities, this thesis gives some researches.In chapter 1, we mainly give the concepts of the sum of distance and the mean distance, and then sum up some results on this two quantities at present.In chapter 2, we calculate the mean distance of some kinds of special graphs by some technique chiefly.In chapter 3, we testify that the conjecture in [40] is wrong by giving an inverse example, and put forward another conjecture.