The Elegant Algorithm Of Random Graph And Its Application In Anti-counterfeiting Of QR Codes

Graph theory is a relatively old branch of mathematics,it is a part of applied mathematics.In history,graph theory has been independently established by Multiple mathematician.The literary record of graph theory first appeared in the 1736 study of Leornhard Euler,who solved the problem of Konigsberg by means of graph theory.Thus,the theory of graphs was born,and Euler was also recognized as the founder of graph theory.In the subsequent development process,many famous conjectures have emerged,such as the Hamiltonian circuit,the four-color conjecture,the graceful tree conjecture,etc.Many problems in reality can be transformed into the related problems of graph theory,which has attracted the attention and extensiveness of scholars.Research has promoted the flourishing of multiple disciplines.In order to solve the relevant conjectures in the graph theory,scholars have conducted extensive research on the graph labeling,and the graph labeling has become a very active branch in the current graph theory.The graph labeling originally originated from the graceful tree conjecture proposed by Rosa in 1966:every tree is a graceful tree.Although the research results are quite numerous,the conjecture has not been resolved so far.As a result,the study of graph labeling has been widely extended,and various types of labels have appeared.The elegant labeling is one of the graph labeling and is the main research object of this thesis.Its concept was first proposed by Chang,Hsu and Rogers in 1981 and gave a clear definition.graph labeling are widely used in computer science,networking,organic chemistry,machine learning and many other fields.Since the number of graphs increases with the increase of the number of points,the label of the graph is manually operated before the computer is used.Therefore,the scope of the scholar's research is relatively simple,mainly limited to the special graph or each type of graph,and the graph is not comprehensive.The study thus does not have a comprehensive understanding of the elegance of the graph.This thesis designs a recursive backtracking algorithm to have a comprehensive study on the elegance of the graph,which has the high efficiency and long-term processing ability of the computer,and combines the pruning and the pre-judgment function.The connected graph is elegantly verified,and all elegant and non-elegant graphs within 9 points are obtained.Due to the limited configuration of computer hardware devices,as the number of graphics increases,the processing time of the computer increases.The tree,single circle graph and bicyclic graphs within 16 points are selected for elegant verification.The obtained results are analyzed,and the relevant conclusions are drawn.According to the conclusions,relevant conjectures are proposed.Based on the conjectures,the elegance of the large-point graphs is judged.By judging the large-point graphs,it is judged whether the conjecture is established.Studies have shown that for graphs,when2?p?9,p?q?2p and q?1?mod4?,all graphs are elegant.when2?p?16,all the odd trees were elegant trees and all the even numbers were almost elegant trees.when2?p?16,all the Single circle graphs were elegant except forC₅?₉C andC_{1 3}.For the graph?p,p?10??1,when4?p?16and p?10?1?28?1?mod4?,the bicyclic graphsC_?m,n?is a non-elegant graph.The odd elegance of the above graphics was verified one by one.Experiments show when2?p?16,all the trees were odd elegant trees.when2?p?9,except for a few graphs,all the graphs with only even circles were odd elegant.QR code has the characteristics of large information capacity,high reliability,low cost and legibility,and it is widely used in various fields of society.In this thesis,the QR code is applied to the anti-counterfeiting aspect of the product,and the elegant matrix of the graph is generated according to the elegant labeling algorithm of the graph,and the"electronic identity code"is created by combining the relevant information of the product.The graph labeling is used as an“electronic identity code”,which is added to the QR code,so that the feedback information can be feedback according to the customer.The authenticity of the goods protects the vital interests of the customers.