Edge-magic total labeling of two joined graphs

If each vertex in a connected graph is labeled such that the values correspond one-to-one with the natural number set from 1 to the sum of vertices and edges, and if the sum of labels of each edge and its two vertices equals a fixed constant, then the connected graph is said to have an edge magic total labeling. In this paper, a new algorithm optimization strategy is designed by combining recursive algorithms with objective functions, achieving the determination of whether all simple connected graphs within a finite set of points conform to edge magic total labeling. Subsequently, through analysis and research by peers, it is found that certain types of graphs are either edge magic total labeling graphs or non-edge magic total labeling graphs when certain conditions are met regarding the number of vertices and edges. Combining existing results, several rules regarding edge magic total labeling are derived, and corresponding theorems are summarized and proven.