The Edge-balance Index Sets Of The Nested Graph With The In（?）nite Paths And Power Circles C_n^m× P_mn（n = 5, 10）

The edge-balance index sets is a signi?cant branch of the Boolean index set which is an important subject of graph labeling. Based on the researches about smaller powercycle nested network graph by Hongjuan Tian, Yanming Chang and Qingwen Zhang,the edge-balanced index sets of ten-power-cycle and ten-power-cycle nested graph were investigated. When n is an odd or a large even number, a new method of changing index is provided, simplifying the proving process. It reduced the diffculty labeling because of the novel design of the basic graph, the results show that the edge-balanced index sets of C_n^m × P_mn（n = 5, m ≥ 4; n = 10, m ≥ 2）. This paper has proved the existence of the edge-balanced index sets of two classes of nested graph, the computational formulas and the construction of the corresponding graphs are also provided.This paper consists of four chapters.In chapter 1, we mainly introduce the background of the labeling problem, the formation and development of edge-balance index sets, and provide some de?nitions needed by this paper.In chapter 2, a new concept of ?ve-vertex sector subgraph-group was proposed.For different m, we use different methods to study its nested-cycles’ edge-index. Using the research methods and techniques of graph theory and combinatorial mathematics,through studying the new design idea about the basic graph, nested-cycle subgraph with gear, the edge-balanced index sets of the power-cycle nested graph C_5^m × P_m5（m ≥ 4）are provided here, and the proofs of the computational of formulas and the construction of the corresponding graphs also give out.In chapter 3, as the number m increases, the diffculty grows by power, when n is 10. It reduced the diffculty labeling because of the novel design of the basic graph,nested-cycle subgraph with gear and single-point sector subgraph. The results show that the edge-balance index sets of C_10^m ×P_m10（m ≥ 2）, and showing the proofs of the computational formulas and the construction of the corresponding graphs.In chapter 4, this part is the conclusion and prospect of the thesis.