A graph labeling is an assignment of integers to the vertices or edges, or both, subject to certain conditions. Graph labelings were first introduced in the late 1960s. In the intervening years dozens of graph labelings techniques have been studied in over 1000 papers. Finding out what has been done for any particular kind of labeling and keeping up with new discoveries is difficult because of the sheer number of papers and because many of the papers have appeared in journals that are not widely available. In this survey I have collected everything I could find on graph labeling. For the convenience of the reader the survey includes a detailed table of contents and index.

/pdf/a-dynamic-survey-of-graph-labeling-198znmwxtq.pdf

A Dynamic Survey of Graph Labeling

Studies in graph theory

Graphical passwords (GPWs) are convenient for mobile equipments with touch screen. Topological graphic passwords (Topsnut-gpws) can be saved in computer by classical matrices and run quickly than the existing GPWs. We research Topsnut-gpws by the matching of view, since they have many advantages. We discuss: configuration matching partition, coloring/labelling matching partition, set matching partition, matching chain, etc. And, we introduce new graph labellings for enriching Topsnut-matchings and show that these labellings can be realized for trees or spanning trees of networks. In theoretical works we explore Graph Labelling Analysis, and show that every graph admits our extremal labellings and set-type labellings in graph theory. Many of the graph labellings mentioned are related with problems of set matching partitions to number theory, and yield new objects and new problems to graph theory.

Topological Graphic Passwords And Their Matchings Towards Cryptography.

Lattice-based Cryptography is considered to have the characteristics of classical computers and quantum attack resistance. We will design various graphic lattices and matrix lattices based on knowledge of graph theory and topological coding, since many problems of graph theory can be expressed or illustrated by (colored) star-graphic lattices. A new pair of the leaf-splitting operation and the leaf-coinciding operation will be introduced, and we combine graph colorings and graph labellings to design particular proper total colorings as tools to build up various graphic lattices, graph homomorphism lattice, graphic group lattices and Topcode-matrix lattices. Graphic group lattices and (directed) Topcode-matrix lattices enable us to build up connections between traditional lattices and graphic lattices. We present mathematical problems encountered in researching graphic lattices, some problems are: Tree topological authentication, Decompose graphs into Hanzi-graphs, Number String Decomposition Problem, $(p,s)$-gracefully total numbers.

Graphic Lattices and Matrix Lattices Of Topological Coding.

A mathematical topology with matrix is a natural representation of a coding relational structure that is found in many fields of the world. Matrices are very important in computation of real applications, s ce matrices are easy saved in computer and run quickly, as well as matrices are convenient to deal with communities of current networks, such as Laplacian matrices, adjacent matrices in graph theory. Motivated from convenient, useful and powerful matrices used in computation and investigation of today's networks, we have introduced Topcode-matrices, which are matrices of order $3\times q$ and differ from popular matrices applied in linear algebra and computer science. Topcode-matrices can use numbers, letters, Chinese characters, sets, graphs, algebraic groups \emph{etc.} as their elements. One important thing is that Topcode-matrices of numbers can derive easily number strings, since number strings are text-based passwords used in information security. Topcode-matrices can be used to describe topological graphic passwords (Topsnut-gpws) used in information security and graph connected properties for solving some problems coming in the investigation of Graph Networks and Graph Neural Networks proposed by GoogleBrain and DeepMind. Our topics, in this article, are: Topsnut-matrices, Topcode-matrices, Hanzi-matrices, adjacency ve-value matrices and pan-Topcode-matrices, and some connections between these Topcode-matrices will be proven. We will discuss algebraic groups obtained from the above matrices, graph groups, graph networking groups and number string groups for encrypting different communities of dynamic networks. The operations and results on our matrices help us to set up our overall security mechanism to protect networks.

pdf/topological-coding-and-topological-matrices-toward-network-4gxdab9ve5.pdf

Topological Coding and Topological Matrices Toward Network Overall Security

Graphical passwords are possibly alternative for text-based passwords The idea of “graphical structure plus number theory” (GSpNT) for creating new type of graphical passwords has been investigated, since the new graphical passwords made by GSpNT needs less storage and implements quickly in network communication We try to find some connections between new graphical passwords defined on a topological structure, and show some them can form algebraic groups in this article By the way, we find new graph labellings in which some mathematical conjectures are produced

On coloring/labelling graphical groups for creating new graphical passwords

Graphical passwords have been applied in information networks, such as QR code and those passwords for opening mobile phones. Topological graphic passwords (Topsnut-GPWs) differ from the existing graphical passwords, since they can be saved in computer by matrices and run quickly than the existing graphical passwords. We show that Topsnut-GPWs can be easily connected with text-based passwords, but no inverse process rewrite a Topsnut-GPW by the text-based passwords produced from this Topsnut-GPW. Our techniques can be associated with some well-known traditional encryptions. We point that Topsnut-GPWs have provable security and computational security since some graph labellings, which make Topsnut-GPWs, are related with mathematical conjectures, and many spaces of Topsnut-GPWs have huge cardinalities. The individuation of Topsnut-GPWs is an important working direction for withstanding breaks and attacks, and approximating unconditional security.

Connection Between Text-based Passwords and Topological Graphic Passwords

Topological graphic passwords (Topsnut-gpws) are one of graph-type passwords, but differ from the existing graphical passwords, since Topsnut-gpws are saved in computer by algebraic matrices We focus on the transformation between text-based passwords (TB-paws) and Topsnut-gpws in this article Several methods for generating TB-paws from Topsnut-gpws are introduced; these methods are based on topological structures and graph coloring/labellings, such that authentications must have two steps: one is topological structure authentication, and another is text-based authentication Four basic topological structure authentications are introduced and many text-based authentications follow Topsnut-gpws Our methods are based on algebraic, number theory and graph theory, many of them can be transformed into polynomial algorithms A new type of matrices for describing Topsnut-gpws is created here, and such matrices can produce TB-paws in complex forms and longer bytes Estimating the space of TB-paws made by Topsnut-gpws is very important for application We propose to encrypt dynamic networks and try to face: (1) thousands of nodes and links of dynamic networks; (2) large numbers of Topsnut-gpws generated by machines rather than human's hands As a try, we apply spanning trees of dynamic networks and graphic groups (Topsnut-groups) to approximate the solutions of these two problems We present some unknown problems in the end of the article for further research

pdf/text-based-passwords-generated-from-topological-graphic-fjhwysmvum.pdf

Text-based Passwords Generated From Topological Graphic Passwords

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