Research On Wiener Index

In recent years,with the application of structure graph theory and topological graph theory in computer network,image processing and other fields,more and more branch theories of graph theory have been studied and expanded,and the bottlenecks of various application also give rise to more practical and rigorous theorem conclusions.In the field of layout design such as base station distribution topology network and social relationship network,cost has been in-depth studied as an important indicator to measure efficiency.Wiener index is one of the most classic and widely used indicators in topology,reflecting the average distance of any node pair in the graph,and greatly reducing the network layout consumption by reducing the Wiener index of the graph.The core problem of this paper is to characterize the structure of Wiener index extremum graph of a class of network topology graphs under given conditions.Research object of the paper is defined as T-unicyclic-graph U(Δ,g,n)or U(Δ,n1,n2,…,ng)with order n,girth g≥3,maximum degree Δ≥4,and its Thanging-tree Ti,where |Ti|=ni,1 ≤ g,and characterize the graph structure when the Wiener index takes extremum.The main research contents of this paper are:(1)Give the structural relationship proof of the minimum Wiener index of the hanging tree in a simple undirected graph,and demonstrate the structural characteristics of the graph of the hanging tree through analysis and calculation.This approach is a prerequisite for the unicyclic graph point movement and graph conversion method below,and is also a key step in the graph structure description.(2)Based on the above important conclusions,the drawing method of branch classification is designed to simply and skillfully compare the difference of the T-hanging-tree Wiener index given the order of each hanging tree and the maximum degree,and the vertex structure and the limit condition are obtained when the T-unicyclic-graph takes extreme Wiener index.Taking the T-unicyclic-graph with girth of 3 and maximum degree not less than 4 as a special case,the difference part of the calculation is divided by the method of branch classification,then the extreme value structure that strictly limits the number of vertices is described.(3)According to the above important conclusions,the graph transformation method of last layer division transfer is designed,and the T-unicyclic-graph is classified and divided into four graph transformation modes.By comparing the Wiener index changes step by step,the vertex structure and its layer number relationship are described when the Wiener index of Tunicyclic-graph given total order is taken to the extreme value.