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.