Most application systems still rely on traditional symbolic password mechanisms for identity authentication. However, frequent password leakage incidents continue to threaten user information security. Although numerous solutions have been proposed, most fail to effectively address the real-time detection of password leaks. This paper innovatively proposes a Pythagorean triple-based password leakage detection method, achieving security protection through a honeywords generation algorithm with high smoothness characteristics. First, we construct a special graph structure sequence by leveraging the number-theoretic properties of Pythagorean triples. Second, based on this sequence, we design a honeywords generation algorithm with an optimal flatness distribution. Third, the security proof verifies the effectiveness of the scheme in resisting brute-force attacks and detecting leaks. Finally, comparative experiments demonstrate that our solution exhibits significant advantages in key metrics such as honeywords flatness, resistance to DoS attacks, and storage efficiency, making it particularly suitable for large-scale user authentication systems.

The development of modern cryptography and related mathematical theories has highlighted lattice-based cryptography as a promising quantum-resistant approach. To advance the study of topological coding, we introduce new techniques based on super total graceful-type labelings/colorings and parameterized labelings/colorings. We propose a randomly-leaf-adding algorithm for finding connections between super total graceful-type labelings and colorings, which can be utilized in the construction of graph lattices. For building graph lattices in a randomized manner, we propose an algorithm that relies on the random addition of leaves. We design NBS algorithms for implementing topological graph cryptography in encrypting digital files. (c) 2025 Elsevier B.V. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

ObjectiveWith the rapid growth of e-commerce, express delivery volumes have surged, placing increased demands on existing logistics infrastructure and operational models. An efficient express logistics network can help reduce costs, improve transportation efficiency, and enhance logistics management. Therefore, analyzing the structure and operation of express logistics networks, as well as identifying ways to optimize these networks, has become a critical focus for logistics companies. The goal is to improve operational efficiency and support balanced regional economic development. Current research on express logistics networks involves constructing various models, such as mathematical optimization models, decision models, and network evaluation models, and applying algorithms like heuristic, genetic, and greedy algorithms, as well as those based on complex network theory, to optimize network structure, performance, and planning decisions. However, a limitation of existing studies is the lack of models closely aligned with the practical realities of express logistics, and the absence of effective new algorithms to address the complex, evolving challenges faced by express logistics networks. This study proposes a novel discrete mathematical model, also known as a topology model, for express logistics networks from the perspective of graph theory. The model comprises a road network (physical network), a topology network (mathematical model), and an information network (soft control system), providing a closer alignment with real-world express logistics scenarios. Through both qualitative and quantitative analyses of the model, along with the design of corresponding optimization algorithms, this research offers a reference for the in-depth study and scientific optimization of express logistics networks.MethodsThis study employs various methods: (1) Mathematical Model Construction: A new discrete mathematical model for express logistics networks is developed, accounting for the nonlinear, stochastic, and discrete characteristics of the network. The model integrates physical, topological, and informational networks. (2) Qualitative Analysis: The topology model of the express logistics network is qualitatively analyzed using graph theory concepts and algorithms, where the network topology model is represented as a weighted structure in graph theory. (3) Quantitative Analysis: The mathematical model is analyzed quantitatively using statistical parameters, optimization algorithms, and other mathematical techniques. The edges in the topological model are assigned route length weights, and new optimization algorithms—such as the distribution algorithm, control set algorithm, and pre-designated subgraph algorithm—are proposed to optimize the express logistics network topology. (4) Case Study and Optimization: The topology model is applied to the express logistics network in the central district of Lanzhou City (Chengguan District), where corresponding optimization algorithms are implemented. Solutions to challenges, such as the computational complexity of the model, are proposed.Results and DiscussionsThe mathematical model in this study is a topological graph based on graph theory, where various matrices are used to input the express logistics network’s topology into the computer for subsequent calculations. Innovation 1: A topological model of the express logistics network is created. Innovation 2: The topological model of the express logistics network is optimized and quantitatively analyzed, and a minimum weight path m-control set algorithm (m ≥ 2) and a pre-designated subgraph control algorithm are developed(Algorithm 3, Algorithm 4). These models and algorithms are then applied to the study of the express logistics network in the Chengguan District of Lanzhou City. Innovation 3: In response to the large-scale data and the limitations in computer computing power, as well as the absence of a super-large computer at the author’s institution, the large-scale matrix calculation is divided into smaller regional matrices for optimized computation. Innovation 4: Different optimization algorithms are selected for different areas of the road network map of Lanzhou City’s Chengguan District (Fig. 4, Fig. 5). Multiple calculation results are integrated to obtain the minimum weight path of the pre-designated subgraph for the Chengguan District, validating the effectiveness of the model and algorithms.ConclusionsThis study addresses existing issues in express logistics network research through the aforementioned work and innovations. A new model and new algorithms, better suited to practical logistics scenarios, are developed. Based on the case study, new problems and methods are proposed, offering further possibilities for optimizing express logistics networks. With the rapid development of emerging technologies such as the Internet of Things, big data, and artificial intelligence, future research could focus on deeply integrating these technologies to enable real-time and accurate collection and analysis of logistics data. Access to high-quality and diverse data can further improve the accuracy of the model’s calculations and enhance intelligent decision-making capabilities. This study has only considered assigning route length weights to the edges in the topological model; future work may explore multi-objective, multi-weight optimization models for express logistics networks to meet the practical decision-making needs of different logistics service providers.

The graph lattice in topological coding is a new encryption technology, like lattice base encryption. The graphs in a graph lattice are stored by matrices and run in a computer. The main theoretical techniques for topological coding come from mathematical disciplines such as discrete mathematics, number theory, and algebra. Because topological coding involves a large number of mathematical conjectures and NP problems, the numeric based strings generated by topological coding have irreversibility and computational security. In this paper, we have show some new labels and coloring, which are defined as follows: k-module edge-difference total coloring, super k-module edge-difference total coloring, set-ordered k-module edge-difference total colorings and edge-difference (k,d) total colorings. We have proven the graph obtained by three kinds of graph operations has the edge-difference type colorings and show the leaf adding algorithm. Our results can proved by algorithmic proofs.

For real application and theoretical investigation of ordinary hypergraphs and non-ordinary hypergraphs, researchers need to establish standard rules and feasible operating methods. We propose a visualization tool for investigating hypergraphs by means of the natural topological structure of finite sets and their subsets, so we are able to construct various non-ordinary hypergraphs, and to reveal topological properties (such as hamiltonian cycles, maximal planar graphs), colorings, connectivity, hypergraph group, isomorphism and homomorphism of hypergraphs.

Lattice cryptosystem is a kind of public key cryptosystem which is resistant to quantum computing attacks. Cryptanalysis is involved in the study of lattice cryptanalysis. There are many problems such as the interdisciplinary characteristics are obvious and the research methods tend to be diversified. This paper defines new graph total colorings: graceful-difference total coloring, odd-edge graceful difference total coloring. It is proven that (1) if a connected bipartite graph [Formula: see text] recognizes ordered graceful-difference labeling, the bipartite graph [Formula: see text] obtained by adding [Formula: see text] leaves to connected bipartite graph [Formula: see text] recognizes a graceful-difference total coloring, (2) in order to establish a random coloring graphic lattice, two operations are given to design the graphic lattice. The proof of the theorem can be transformed into a feasible and effective algorithm.

The security of traditional public key cryptography algorithms depends on the difficulty of the underlying mathematical problems. Asymmetric topological encryption is a graph-dependent encryption algorithm produced to resist attacks by quantum computers on these mathematical problems. The security of this encryption algorithm depends on two types of NP-complete problems: subgraph isomorphism and graph coloring. Topological coding technology refers to the technology of generating key strings or topology signature strings through topological coding graphs. We take odd-graceful labeling and set-ordered odd-graceful labeling as limiting functions, and propose two kinds of topological coding generation technique, which we call the random leaf-adding operation and randomly adding edge-removing operation. Through these two techniques, graphs of the same scale and larger scales can be generated with the same type of labeling so as to derive more number strings, expand the key space, and analyze the topology and property of the generated graphs.

With the fast development of quantum computer technology, one has to focus on the security of information running in real networks. Topological coding is a technology of basic topological structure, number theory and algebra has potential application value in asymmetric encryption system. The public key and private key of topological graph encryption system designed by topological coding may be able to resist attacks equipped with artificial intelligence technology and quantum computer. In order to further investigate topology encoding, we propose new techniques for designing public and private keys using graph coloring and labeling.

Let $ f $ be a set-ordered edge-magic labeling of a graph $ G $ from $ V(G) $ and $ E(G) $ to $ [0, p-1] $ and $ [1, p-1] $, respectively; it also satisfies the following conditions: $ |f(V(G))| = p $, $ \max f(X) < \min f(Y) $, and $ f(x)+f(y)+f(xy) = C $ for each edge $ xy\in E(G) $. In this paper, we removed the restriction that the labeling of vertices could not be repeated, and presented the concept of magical colorings including edge-magic coloring, edge-difference coloring, felicitous-difference coloring, and graceful-difference coloring. We studied the magical colorings on the tree and proved the existence of four kinds of magical colorings on the tree from a set-ordered edge-magic labeling. Further, we revealed the transformation relationship between these kinds of colorings.

With the rapid development of quantum computing theory and quantum computer technology, the traditional public-key cryptographic algorithms face great challenges. A technology that might be able to resist attacks equipped with AI technique and quantum computers is the topological graphic password of topological coding. In order to further study topological coding, we use the multiple constraints of graph colorings and labelings to propose new techniques made by felicitous-type labelings and parameterizing felicitous-type labelings of graph theory. We show the RANDOMLY-LEAF-adding algorithm and some connections between felicitous-type labelings and parameterizing felicitous-type labelings since they can be applied to building up graphic lattices.

Abstract We use topological graphs to generate topological coding, which is a fundamental technology in number theory, algebra, and has potential application value in asymmetric encryption systems. This article proposes a method of using topological graph sequences to generate topological coding for protecting hash ciphers, which can effectively protect hash ciphers even in the event of password leakage. We use topology graphs with labelings and colorings to generate graph lattice, which is the application of graph markers in asymmetric encryption systems. There are various kinds of graph labeling, one of them is a harmonious labeling. In order to further study topological coding, we use the graph harmonious-type labelings and colorings to propose (k, d)-harmonious labelling , since they can be applied to build up a new approach of asymmetric encryption. Finally, several typical attack methods about graph lattice analyzed to verify the effectiveness of our scheme.

The graph connectivity is a fundamental concept in graph theory. In particular, it plays a vital role in applications related to the modern interconnection graphs, e.g., it can be used to measure the vulnerability of the corresponding graph, and is an important metric for reliability and fault tolerance of the graph. Here, firstly, we introduce two types of divided operations, named vertex-divided operation and edge-divided operation, respectively, as well as their inverse operations vertex-coincident operation and edge-coincident operation, to find some methods for splitting vertices of graphs. Secondly, we define a new connectivity, which can be referred to as divided connectivity, which differs from traditional connectivity, and present an equivalence relationship between traditional connectivity and our divided connectivity. Afterwards, we explore the structures of graphs based on the vertex-divided connectivity. Then, as an application of our divided operations, we show some necessary and sufficient conditions for a graph to be an Euler's graph. Finally, we propose some valuable and meaningful problems for further research.

给出树的邻和可区别 2-全染色方案,并结合三正则图最小消圈集的独立性以及消圈子图的无圈性,较为简洁地证明三正则图的邻和可区别全色数满足 1-2猜想.进一步利用独立消圈集法确定r-正则图、Halin图以及路与路的笛卡尔乘积图的邻和可区别全色数.

Using asymmetric topology cryptography to encrypt networks on the basis of topology coding is a new topic of cryptography, which consists of two major elements, i.e., topological structures and mathematical constraints. The topological signature of asymmetric topology cryptography is stored in the computer by matrices that can produce number-based strings for application. By means of algebra, we introduce every-zero mixed graphic groups, graphic lattices, and various graph-type homomorphisms and graphic lattices based on mixed graphic groups into cloud computing technology. The whole network encryption will be realized by various graphic groups.

The security of passwords generated by the graphic lattices is based on the difficulty of the graph isomorphism, graceful tree conjecture, and total coloring conjecture. A graphic lattice is generated by a graphic base and graphical operations, where a graphic base is a group of disjointed, connected graphs holding linearly independent properties. We study the existence of graphic bases with odd-graceful total colorings and show graphic lattices by vertex-overlapping and edge-joining operations; we prove that these graphic lattices are closed to the odd-graceful total coloring.

The coming quantum computation is forcing us to reexamine the cryptosystems people use. We are applying graph colorings of topological coding to modern information security and future cryptography against supercomputer and quantum computer attacks in the near future. Many of techniques introduced here are associated with many mathematical conjecture and NP-problems. We will introduce a group of W-constraint (k,d)-total colorings and algorithms for realizing these colorings in some kinds of graphs, which are used to make quickly public-keys and private-keys with anti-quantum computing, these (k,d)-total colorings are: graceful (k,d)-total colorings, harmonious (k,d)-total colorings, (k,d)-edge-magic total colorings, (k,d)-graceful-difference total colorings and (k,d)-felicitous-difference total colorings. One of useful tools we used is called Topcode-matrix with elements can be all sorts of things, for example, sets, graphs, number-based strings. Most of parameterized graphic colorings/labelings are defined by Topcode-matrix algebra here. From the application point of view, many of our coloring techniques are given by algorithms and easily converted into programs.

设f:V(G)∪E(G)→[1,k]是图G的一个非正常k-全染色.令φ(x)=f(x)+∑e?xf(e)+∑y∈N(x)f(y),其中N(x)={y∈V(G)|xy∈E(G)}.对任意的边uv∈E(G),如果有φ(u)≠φ(v)成立,则称f是图G的一个邻点全和可区别(简记NFSD)k-全染色.图G的邻点全和可区别全染色中最小的k值称为G的邻点全和可区别全色数,记为fgndiΣ(G).通过构造染色函数法,确定了广义Petersen图和循环图的邻点全和可区别全色数.

数学的拓扑图可以自然地表示编码关系结构,也叫做拓扑图编码,这种关系结构在许多领域里得到应用.本文将图的全着色和图的边魔幻标号结合产生特殊的全着色,边魔幻tcn-纯全着色和均匀魔幻tcn-纯全着色.研究了树的边魔幻tcn-纯全着色,均匀魔幻tcn-纯全着色,以及具有极值性质的边魔幻全着色数,确定了特殊全着色在树上的精确着色数,并指出可以推广到含圈图上去.

Let f: V(G)∪ E(G)→{1,2,…,k} be a non-proper total k-coloring of G. Define a weight function on total coloring as ϕ(x)=f(x)+∑_e∋ xf(e)+∑_y∈ N(x)f(y), where N(x)={y∈ V(G)|xy∈ E(G)}. If ϕ(x)≠ϕ(y) for any edge xy∈ E(G), then f is called a neighbor full sum distinguishing total k-coloring of G. The smallest value k for which G has such a coloring is called the neighbor full sum distinguishing total chromatic number of G and denoted by fgndi_∑(G). The coloring is an extension of neighbor sum distinguishing non-proper total coloring. In this paper we conjecture that fgndi_∑(G)≤ 3 for any connected graph G of order at least three. We prove that the conjecture is true for (i) paths and cycles; (ii) 3-regular graphs and (iii) stars, complete graphs, trees, hypercubes, bipartite graphs and complete r-partite graphs. In particular, complete graphs can achieve the upper bound for the above conjecture.

Characterizing the topology and random walk of a random network is difficult because the connections in the network are uncertain. We propose a class of the generalized weighted Koch network by replacing the triangles in the traditional Koch network with a graph Rs according to probability 0≤p≤1 and assign weight to the network. Then, we determine the range of several indicators that can characterize the topological properties of generalized weighted Koch networks by examining the two models under extreme conditions, p=0 and p=1, including average degree, degree distribution, clustering coefficient, diameter, and average weighted shortest path. In addition, we give a lower bound on the average trapping time (ATT) in the trapping problem of generalized weighted Koch networks and also reveal the linear, super-linear, and sub-linear relationships between ATT and the number of nodes in the network.