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Based on the graph generation algorithm, we propose a novel algorithm to solve the problem of D(β)-vertices sum distinguishing edge coloring of some kinds of special graphs. Our main contributions to solving this graph coloring problem are: firstly, we generate the adjacent matrices of the following graphs; secondly, we input these matrices to our designed algorithm; thirdly, the algorithm will output the corresponding D(β)-vertices sum distinguishing edge coloring matrices; fourthly, through analyzing of the results obtained by our algorithm, we conclude that regular chromatic number are found in some of the graphs involved. We think that our proposed algorithm is innovative and can be adopted by other researchers.

If the labeling values of edges and vertexs for a graph $G(p, q)$ are mapped to $\{1, 2,, p+q\}$ , and the sum of the labels of any two vertexs and their associated edges is equal to a constant, this labeling is called edge-magic total labeling. In this paper, a recursive algorithm is designed to obtain the edge-magic total labeling of the finite vertex composite graph. Through the analysis of the results, It is found that the composite graph $F_{n}\rightarrow C_{l}\uparrow S_{m}$ is an $EMTL$ graph, and then some theorems are summarized and proved.

We propose a new algorithm, based on the graph generation method, to solve the problem of super edge-magic total labeling of graphs with a large number of vertices. Our main contributions to solving the graph labeling problem are: first, we generate the adjacent matrices of graphs composed of fans and stars. Second, we input these matrices to our proposed algorithm. Third, if the labeling is successful, the algorithm will output the corresponding super edge-magic total labeling matrices. Otherwise, no super edge-magic total labeling exists for the graphs involved. Forth, through the analysis of the results obtained by our algorithm, we conclude that regular labels are found in some of the graphs involved. Our algorithm can distinguish between graphs that have a super edge-magic total labeling and graphs that don't efficiently. We think that our proposed algorithm is innovative and can be adopted by other researchers.

A new algorithm for vertex-distinguishing edge coloring of graph is designed. First of all, by definition of vertex-distinguishing edge coloring, we know that there exist two constraints rules, one is the edge constraint and the other is the vertex color set constraint. Second, we established objective function according to the two constraint rules. Third, based on the exchange rules, an algorithm is designed to make the objective function meet the requirements. The algorithm steps are described in detail. The experimental results show that the vertex-distinguishing edge coloring obtained by the algorithm conforms to the coloring conditions. This algorithm can effectively solve this coloring problem of graphs with large number of vertices.