The Research Of Algorithms And Application Of Multiple Graceful Labellings And Edge-magic Labelling

The graph labelling is a branch of the study of graph theory.It was proposed by Rosa et al.in 1966 to solve Ringel's conjecture.The graph labelling is the mapping of the vertex set and the edge set of the graph to the integer set.According to the different requirements of the edge labelling,various types of labellings are generated,such as: graceful labelling,odd-graceful labelling,felicitous labelling,odd-elegant labelling and edge-magic labelling and so on.Since the graph labelling is widely used in many fields such as complex networks,big data,computer theory,operations research,organic chemistry,systems science,and graphical password,many researchers have done a lot of work in this area,but there are still many problems that have not been resolved in graph labelling.Some conjectures have not been proved or denied so far.For example,all trees are graceful,all trees are magical,and all trees are odd-graceful.In the existing literature,graph labelling of is to describe the graph with mathematical formulas,and then the combination constructs to prove that the graphs have labellings.Because of the difficulty in describing the random graphs,the main research is the special features that are easy to describe.Graph or a certain type of graphs,such as caterpillar trees,lobster trees,star graphs,fan graphs,wheel graphs,circle graphs,joint graphs,etc.This kind of research method is almost a tentative means,that is,finding the law of labelling in a large number of very special cases,and finding the path of proof.This method has a large workload,a cumbersome content,a special model,low algorithm efficiency,and poor flexibility,so there is no way to prove whether a simple connected graph has a labelling.In this paper,the computer algorithm is used to solve the labelling problem of random graphs.The graph generation algorithm is used to generate the required the library of non-isomorphic graphs.By analyzing the constraints of several graceful labelling and edge-magic labelling and the corresponding label features,the corresponding value space is calculated,design labelling algorithm.Applying the obtained labelling algorithm to the graphic password to solve the reality problem,the main research work of this paper is as follows:(1)Introducing the related concepts of the graph,the existing scheme of the authentication scheme of the graphic password,and the existing conclusion of the magical total labelling.(2)Designing and implementing several graceful labelling algorithms for hierarchical-graph.Firstly,the properties of the hierarchical-graph are proved.The combined construction method and the algorithm implement the set-ordered graceful labellings of hierarchical-graph.Secondly,based on the hierarchical-graph,the graph is formed by adding a leaf to each vertex,and the algorithm realizes that the graph has strong graceful labellings.Finally,based on the hierarchical-graph,the graph formed by arbitrarily adding leaf node to each vertex proves that the graph has odd-graceful labellings.(3)Designing and implementing the edge-magic total labelling algorithm of the graphs.There are two main algorithms,one of which is the super edge-magic total labelling algorithm of random graphs.At present,the super edge-magic labelling of all random graphs within 9vertice can be realized,and the super edge-magic graph and the non-super edge-magic graph are obtained,and the experimental results are summarized and the theorem is obtained: all the tree graphs within 9 vertice are super edge-magic graphs.The relevant conclusions of the other graphs within 9 vertice are obtained.The other is the edge-magic total labelling algorithm of tree graphs,unicyclic graphs and bicyclic graphs.At present,the edge-magic labelling of the tree graphs,the unicyclic graphs and the bicyclic graphs of 10 vertice can be realized.The experimental results of the algorithm are summarized and analyzed and the theorem is obtained: all the tree graphs within 10 vertice are edge-magic graphs,all unicyclic graphs within 10 vertice are edge-magic graphs,and all bicyclic graphs within 10 vertice are edge-magic graphs.(4)Using the easy-to-remember of the graphics and the above-mentioned existing graph labelling algorithm,the graph labelling is applied to the new graphic password,and a graphic and labelling algorithm comes up,and the feasibility analysis is carried out.