Vertex-magic Total Labeling Algorithm Of The Random Graph And Its Application

As an important branch of graph theory,the history of graph labeling can be traced back to Rosa's " graceful tree conjecture ",which laid a foundation for the development of graph labeling.Later in the process of the study of graph labeling,some conclusions and properties are obtained and widely used in many fields,especially in the field of computer related,which makes the problem of graph labeling be paid much attention.The methods using graph labeling to solve the problem is looking for the graph theory models from actual problems in the implicit and abstract intuitive ways,the study of the graph theory models not only solves the actual problem but also promotes the development of the graph theory itself.Graph labeling,which uses the elements in the integer set to allocate the vertices and edges of the graph,so as to make it meet certain conditions.According to different conditions,scholars have gradually put forward and improved the concepts of various labels and found a way to label different graphs,and finally established a series of labeling theories.There are hundreds of concepts of graphing labeing,such as graceful labeling,magic labeling and so on,and so are the research methods.However,most of the results obtained so far are about special graphs that are easy to describe,such as path,circle,star,fan,wheel,complete bipartite graph and their combination graph,while there are few research results on random graphs.With the development of computer software and hardware,it is a new research method and new thought to use computer technology to design algorithm for random graph to solve the problem of graph labeling.This paper aims at the vertex-magic total labeling algorithm and the(a,d)-vertex-antimagic edge labeling algorithm in the magic labeling,the algorithm is used to obtain the label numbers of the non-isomorphic graphs in the finite points.At the same time,through the analysis of some results obtained,the labeling conclusion of a class of graphs is obtained.The main research work of this paper is as follows:(1)Introducing the related concepts of graph labeling and the research status of vertex-magic total labeling and(a,d)-vertex-antimagic edge labelings,and analyzing the existing problems of these two labelings;(2)Designing and completing a recursive search algorithm based on the search vertex magic solution space.Firstly,relevant concepts and theorems are used to determine the algorithm steps and the traditional labeling method is used to verify its correctness.Secondly,the algorithm is used to obtain the labeling matrix of non-isomorphic composition within 9 points and the adjacency matrix of graphs without labels.After collecting data and analyzing results,the labeling conclusion of a class of graphs is obtained.And on the basis of the optimization of time complexity,designing another two labeling algorithms,respectively are constructed and adjacency matrix method based on combination labeling algorithm.Comparing the test and the comparison of the input data of three kinds of algorithm of,a search based on vertex magic solution space recursive search algorithm is proved to be the most practical.Finally,we designed the super vertex-magic total labeling algorithm and a algorithm aimed at getting all the vertex magic total labeling solutions of a single graph,and through them the labeling results of the graphs are obtained;(3)The design and implementation of(a,d)-vertex-antimagic edge labeling algorithm is finished,and the labeling tuple of the random graph is obtained,by which the labeling is completed in the adjacency matrix.The algorithm is used to realize the labeling of isomorphism graph within 9 points,and the conclusion is drawn that the labeling does not exist in the unicyclic graph.(4)Using the properties of labeling graph and apply them to the graphic password design,a new graphic password scheme is proposed to solve the existing shoulder peeping attack of the graphic password.