Based on the equal-cycle nested graph, the power-cycle nested graph is brought forward. In this paper, we research on the largest edge-balance index of the graph C_(7^m)×P_(m_7) (m≡2(mod3))(m≥5) by the methods and techniques of graph theory and combinatorial mathematics, and solve formula proof and graphic tectonic methods.

Based on the research of Power-cycle Nested Graph C_6^m×P_6^m, the decomposition method of single point sector has come up. By the use of the process of clawed nested-cycle sub-graph, the edge-balanced index sets of the power-cycle nested graph C_6^m×P_6^m are solved when m ≥4 and m=4(mod5) . Besides, the constructive proofs of the computational formulas are also completed. The theory can be applied to information engineering, communication networks, computer science, economic management, medicine, etc. The proving method can be a reference to solve the problem of the power-cycle nested graph C_6^m×P_6^m.

Based on the power-cycle nested graph brought before, 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 and five-vertex sector subgraph-group, the edge-balanced index sets of the power-cycle nested graph are provided here, for and , and the proofs of the computational of formulas and the construction of the corresponding graphs also give out.

The research of Boolean index sets of graphs is one of the most important graph theories in the graph theory . Boolean index sets of graphs are to use the vertex sets and the edge sets of graphs to study the characteristics of various graphs and their inherent characteristics through corresponding the mapping function to Z_2. It’s theory can be applied to information engineering, communication networks, computer science, economic management, medicine, etc. The edge-balance index set is an important issue in Boolean index set. In this paper, we defined edge-friendly labeling of the graph, edge-balance index set of the graph and the graph {[C_(3^m)×P_(m_3)](m≥3)}. we completely determine the edge-balance index sets of the graph [C_(3^m)×P_(m_3)] for the integer [m≡3,0 (mod 4)] and solve formula proof and graphic tectonic methods.

The research of Boolean index sets of graphs is one of the important graph theory in the graph theory. Boolean index sets of graphs are to use the vertex sets and the edge sets of graphs to study the characteristics of various graphs and their inherent characteristics through corresponding the mapping function to Z_2. Its theory can be applied to information engineering, communication networks, computer science, economic management, medicine, etc. The edge-balance index set is an important issue in Boolean index set. In this paper, we defined edge-friendly labeling of the graph, edge-balance index set of the graph and the graph C_(n^m) × P_(m_n) (n=4,m≥2). We completely determine the edge-balance index sets of the graph C_(4^m) × P_(m_4) (m=0(mod3)) and solve formula proof and graphic tectonic methods.