The Edge-balance Index Sets Of The Nested Graph With The Infinite Paths And Power Circles C_6^m×P_m6（m≥2）

Graph theory is a branch of mathematics, which takes graph as the research object,and is an especially important branch of discrete math. The labeling problem of graph is an important research topic in graph theory, and the edge-balance index is one of the hot issues. Numerous domestic and international researchers are working to the research in this aspect and have won a series of research results.Based on the senior sister apprentices’ research about infinite paths power-nested graph with the three-cycle decomposition method, we introduce the concept of the five-cycle decomposition method and single point sector sub-graph. The edge-balanced index sets of the power-cycle nested graphmnmnC ?P are solved when n ?6 and find out the corresponding calculation formula and the construction of edge-balance index set.This paper consists of four chapters:In chapter 1, we introduce the background of the graph labeling problem, the formation and the domestic and foreign research status of the edge-balanced index set.Some definitions needed by this paper are also given in this part.Based on the five-cycle decomposition method and the single point sector sub-graph, we make a smart structure for each types of graph’s labeling function.According to the dual characteristics of the parity and the recursive method of the graph,the edge-balanced index sets of power-nested graph and the construction of the graph is completed in chapter 3 and chapter 4.In chapter 2, the edge-balanced index sets of the power-cycle nested graph? ?662m mC ?P m ? are solved when m ??2,3,4??mod5?In chapter 3, the edge-balanced index sets of the power-cycle nested graph? ?662m mC ?P m ? are solved when m ??5,6??mod5?In chapter 4, Summary and Scope.