Graph product techniques for cordial labeling on cubic root graph structures

The computational complexity of cordial and equitable labelling

<p>The concepts of cordial labeling, signed product cordiality, and logical cordiality have been introduced independently by different researchers as distinct labeling schemes. In this paper, we demonstrate the equivalence of these concepts. Specifically, we prove that a graph $ G $ is cordial if and only if it is signed product cordial, if and only if it is logically cordial. Additionally, we establish that a graph $ G $ admits permuted cordial labeling if and only if it exhibits cubic roots cordial labeling. Furthermore, we leverage this newfound equivalence to analyze the cordiality properties of several standard graphs.</p>

The article analyzes the existing legal mechanisms for managing critical information infrastructure in Ukraine. The instruments for their improvement are proposing in this article. An important component of critical infrastructure is its information component – a critical information infrastructure. The sphere of protection of critical information infrastructure in Ukraine is at the initial stage of formation. The current legislation defines only certain objects of socio-economic sphere, in which extraordinary events can lead to socially dangerous consequences. In view of the fact, that the term “critical information infrastructure” does not having a consistent interpretation in different countries, we propose our opinion. “Critical information infrastructure is a system of information management of critical facilities and information and communication networks that provide defense capabilities and security of public and private institutions, whose operation may flow to the national security of Ukraine” (KII). In the KII we can identified information and network components. Information environment of KII is a system for information management of critical objects, including computing and information resources that form automated control systems (ACS). The network component of KII consists of a set of telecommunication devices, communication lines and network equipment, systems of open protocols for the exchange of information between elecommunication devices, global system of digital addresses and digital identifiers, software. The Internet network can be considered as a technological add-on over a telecommunication network that provides the provision of data transmission and processing services (e-mail, teleconferencing, file transfer, access to computing and information systems in local area networks). The main threat to the safety of ACS of critical information infrastructure objects is targeted actions on information systems, information and telecommunication networks by software and hardware. KII legal security include two main components – national and international. The national component may be forming by a set of principles, legal institutions and norms, which are enshrined in the national legislation regulating public relations in Ukraine in the area of counteracting the security threats of the ACS of critical objects. In order to protect the most important objects of KII, it is necessary to identify these objects. The current legislation defines such categories of objects, for which special conditions for ensuring their protection and functioning are established. Some of them, in whole or in part, may be classifying as objects of critical infrastructure. The specificity of providing information security was reflecting in such Ukraine laws like “On the Fundamentals of National Security of Ukraine”, “On the Concept of the National Program of Informatization”, “On the National Program of Informatization”. As well as the Concept of Development of the Security and Defense Sector of Ukraine, the National Security Strategy of Ukraine, the Strategy of Cybersecurity Of Ukraine. The National Security Strategy identifies actual threats to national security and sets priorities for information security, cyber security and security of information resources and critical infrastructure. At the same time, the implementation of the state policy in the field of security of KII requires the further development of legal principles and norms governing the relevant social relations, that is, the national component of the legal security of KII. Ukraine should ensure the establishment of a nationwide system for assessing risks and threats to critical infrastructure, and after the legislative definition of the main terms, the implementation of the Identification of Critical Information Infrastructure objects. Identification of objects of critical information infrastructure can be accomplishing by introducing the certification of objects of critical information infrastructure. Such passports must contain general data about the facility, data on the main sources of danger, data on hazardous natural conditions, technological processes and response to threats. The international component of the legal security of KII provides for the regulation of a set of principles and norms defined by international treaties and recognized by the state, regulating issues of international cooperation in this area. Ukraine has signed the Convention on Cybercrime together with the member states of the Council of Europe and other States. It is aiming at stopping actions against the confidentiality, integrity and availability of computer systems, networks and computer data, as well as abusing such systems, networks and data by installing the criminal responsibility for such behavior, the provision of powers sufficient to combat criminal offenses, and the conclusion of agreements on rapid and reliable international cooperation. In addition, the plan of measures for 2017 on implementation of the Cybersecurity Strategy of Ukraine provides for the implementation of Directive 2008/114/EC on the protection of critical infrastructure, in particular on cybersecurity and cyber defense of critical infrastructure objects. Development of the system of international information security, the following main groups of international relations that requirenormative legal regulation within the framework of the legal security of KII: definition of the boundaries of the national KII in the global information and communication infrastructure and fixing signs of computer incidents in the control system of critical objects information infrastructure. The absence of generally recognized borders of state sovereignty of States in this space is a significant obstacle to the application of international law to the actions of other states. In particular, this impedes the establishment of limits of responsibility of states for violating the security of the KII and organizing international cooperation in the field of countering computer crime. The urgency of the legislative consolidation of signs of computer incidents in the automated control system of critical information infrastructure objects suggests the widespread use of the concept of “incident” in international law. An incident in cyberspace usually associated with a violation of the functioning of the components of cyberspace – an electronic collection environment and automated processing of information that determines the processes of the implementation of these operations, as well as information systems and automated control systems. The essence of the general definition of the “international incident” in the field of KII will be determining, firstly, by the nature of international relations between states that are violating by the “incident”. This event may be the result of unforeseen actions of the state, including actions that harm the interests of public bodies of one or more states, or, conversely, be one of many intentional but minor provocations carried out by agents of one state against another state. Given that international relations in the field of incidents in the field of KII are not regulating by international treaties, the main and, in fact, the only source of international law in this case serves as an international custom, however, its application to the sphere of KII is accompanying by considerable difficulties. For Ukraine, it is possible to introduce the positive experience of other states in the security of the KII. In particular, the problem of security of information technologies has been enshrined in the international standard ISO / IEC 15408 “General criteria for assessing the safety of information technology”.

The product cordial labeling is a variant of cordial labeling. Here we investigate product cordial labelings for alternate triangular snake and alternate quadrilateral snake graphs.

Let G be a (p, q) graph. Let k be an integer with 2 ≤ k ≤ p and f from V (G) to the set {1, 2, . . . , k} be a map. For each edge uv, assign the label |f(u) − f(v)|. The function f is called a k-difference cordial labeling of G if |νf (i) − vf (j)| ≤ 1 and |ef (0) − ef (1)| ≤ 1 where vf (x) denotes the number of vertices labelled with x (x ∈ {1, 2 . . . , k}), ef (1) and ef (0) respectively denote the number of edges labelled with 1 and not labelled with 1. A graph with a k-difference cordial labeling is called a k-difference cordial graph. In this paper we investigate the 3-difference cordial labeling of wheel, helms, flower graph, sunflower graph, lotus inside a circle, closed helm, and double wheel.

One important problem in graph theory is graph coloring or graph labeling.Labeling problem is a well-studied problem due to its wide applications, especially in frequency assignment in(mobile) communication systems, coding theory, x-ray crystallography, radar, circuit design, etc.For two non-negative integers, labeling of a graph is a function from the node set to the set of non-negative integers.For any two distinct vertices and of , a radio labeling is an injective function : 0 such that , | | 1 where is the diameter of .The radio number of , is the maximum value of taken overall radio labelings of .This paper determines the radio number of double triangular snake graphs.

A graph with a vertex set is said to be a prime graph if there exists a bijective mapping , where denotes the number of vertices in , such that for any two adjacent vertices and in have . Tensor Product graph is a way to combine (compose) two graphs into one larger and more complex graph. The result is a new graph that reflects the connection properties of the two original graphs, but in a very specific and more complex way than other graph operations. Therefore, this research aims to determine whether there is prime labeling in the class of graphs resulting from the Tensor Product of the path graph and the cycle graph . The research employed analytical and exploratory methods with a trial-and-error strategy to determine the labeling that possesses a prime property. The results of this study prove that two classes of the Tensor Product graph for , and graph , for are prime graph. This finding expands the results on classes of graphs that admit prime labeling and provides a basis for further research on graph labeling in other graph operations

This paper analyzes the optimization features of machine learning (ML) model training procedures using multi-GPU systems to enhance cyber security in telecommunication networks. A key aspect of the study is the use of data parallelism, which allows the distribution of the training load across multiple GPUs, significantly reducing training time and improving model accuracy-critical factors for rapid threat detection in cyberspace. A novel approach for optimizing data batch size using Mutual Information (MI) is proposed, which harmonizes the utilization of computational resources with the information content of the training data. MI helps to determine the optimal data batch size that minimizes training errors and improves model accuracy without a significant increase in training time. Experimental results demonstrate the substantial advantages of multi-GPU configurations compared to single-GPU setups, providing faster training and improved model accuracy. It was particularly emphasized that MI-guided batch size tuning significantly outperforms traditional manual tuning methods, ensuring higher validation accuracy and reducing training time. The study showed that the MI-based approach is an effective tool for addressing the problem of optimizing ML model training processes in real-world scenarios where cyber security is critical. The proposed methods allow ML models to train faster and more accurately identify potential threats, making them particularly relevant for telecommunication networks where a rapid response to new threats in real time is required. The implementation of modern computational technologies such as multi-GPU systems and MI-optimized training enhances the efficiency and accuracy of machine learning models. This, in turn, improves cyber security measures and ensures a more reliable defence of telecommunication networks against malicious attacks. It is noted that the proposed approaches can be adapted not only for cyber security but also for other areas where high model accuracy and fast training are important. Future research prospects include the development of new machine learning methods, particularly deep neural networks, the exploration of alternative computational architectures such as quantum computing or distributed systems, and their integration into real-time systems. Special attention should be paid to the ethical aspects of implementing automated cyber security systems, particularly in preventing bias in algorithms and ensuring fairness in their application.

Objective: To consider the main approaches to organization of systems and communication networks on high-speed railway transport and provide cyber security on the basis of current international and home practice in the given sphere. Methods: Game-theory approach, hierarchy analysis method as well as topological transformation of stochastic networks were applied in the given study. Results: It was detected that a single realization of organizational and technical measures on cyber security cannot provide strong security from threats of cyber security during a long period of time; the latter is connected with constant development of data infrastructure, modification of methods, as well as the methods of attacks, potential attacker opportunities and other factors. Telecommunication networks’ security (TKS) of OJSC “Russian railways” and high-speed railway lines (VSM) from cyber impact is a systemic problem, for the solution of which fundamental research studies are to be conducted, aimed at the detection of basic patterns of information confrontation and directions of TKS cyber security provision and, subsequently, research and development activities for creating an automated communication control system of OJSC “Russian railways” and VSM, the constituent part of which is a subsystem of TKS information security. It is advisable to create an expert council based at regional transport universities, which provide extensive analysis conduct of systems and means of providing information security offered by the industry, as well as cyber security of OJSC “Russian railways” and VSM critically important systems including TKS. Practical importance: The realization of the suggested approach to OJSC “Russian railways” and VSM cyber security provision will make it possible to form the conditions for safe functioning of elements of information infrastructure, provide the creation, effective functioning and the improvement of cyber security control system, as well as to minimize damage from possible attack attempts.

A random walk labeling of a graph G is any labeling of G that could have been obtained by performing a random walk on G. Continuing two recent works, we calculate the number of random walk labelings of perfect trees, combs, and double combs, the torus C 2 × Cn , and the graph obtained by connecting three path graphs at their ends.

Let f be a function from V (G) to {0,1,2}. Give a numerical value (label) from {0, 1, 2} for every edge uv. f is called a mean cordial labeling if |v_f (i)-v_f (j)|≤1 and |e_f (i)-e_f (j)|≤1, ∀ i,j ∈ {0,1,2} where v_f (x) and e_f (x) represent the number of vertices and edges respectively labeled as x(x = 0,1,2). A graph with mean cordial labeling is called a mean cordial graph. In this paper, we study the mean cordial labeling pattern of shadow-graphs of path graphs D_2 (P_n ), for n≥2

The fastest deterministic algorithms for connected components take logarithmic time and perform superlinear work on a Parallel Random Access Machine (PRAM). These algorithms maintain a spanning forest by merging and compressing trees, which requires pointer-chasing operations that increase memory access latency and are limited to shared-memory systems. Many of these PRAM algorithms are also very complicated to implement. Another popular method is “leader-contraction” where the challenge is to select a constant fraction of leaders that are adjacent to a constant fraction of non-leaders with high probability, but this can require adding more edges than were in the original graph. Instead we investigate label propagation because it is deterministic, easy to implement, and does not rely on pointer-chasing. Label propagation exchanges representative labels within a component using simple graph traversal, but it is inherently difficult to complete in a sublinear number of steps. We are able to overcome the problems with label propagation for graph connectivity. We introduce a surprisingly simple framework for deterministic, undirected graph connectivity using label propagation that is easily adaptable to many computational models. It achieves logarithmic convergence independently of the number of processors and without increasing the edge count. We employ a novel method of propagating directed edges in alternating direction while performing minimum reduction on vertex labels. We present new algorithms in PRAM, Stream, and MapReduce. Given a simple, undirected graph [Formula: see text] with [Formula: see text] vertices, [Formula: see text] edges, our approach takes O(m) work each step, but we can only prove logarithmic convergence on a path graph. It was conjectured by Liu and Tarjan (2019) to take [Formula: see text] steps or possibly [Formula: see text] steps. Our experiments on a range of difficult graphs also suggest logarithmic convergence. We leave the proof of convergence as an open problem.

This study investigates the existence of discrete and strong discrete labeling for several classes of path and star-related graphs, contributing to the broader field of graph labeling theory. A graph is said to admit a discrete labeling if its vertices can be assigned binary labels (0 and 1), such that each edge receives a label determined by the exclusive OR (XOR) of the labels of its incident vertices, subject to specified constraints. We first establish that the super subdivision of a path <img src=image/13441705_01.gif> by the complete bipartite graph <img src=image/13441705_02.gif> is discrete, demonstrating that replacing each edge of <img src=image/13441705_01.gif> with <img src=image/13441705_02.gif> preserves the discrete labeling property. Additionally, we prove that the square of a path <img src=image/13441705_01.gif>, obtained by connecting vertices at a distance of at most two, also admits a discrete labeling. Further, we examine the shadow graph <img src=image/13441705_04.gif>, constructed by duplicating <img src=image/13441705_01.gif> and connecting each vertex to its copy's neighbors, and show that it is also discrete. Another variant, the graph <img src=image/13441705_03.gif> formed by switching one pendant vertex in <img src=image/13441705_01.gif>, is similarly proven to support discrete labeling. Beyond paths, we explore star-related graphs, confirming that the shadow graph of a star <img src=image/13441705_05.gif> and the shadow graph of a bi-star <img src=image/13441705_06.gif> both permit discrete labeling. Extending the concept, we introduce strong discrete labeling, which imposes the stricter requirement by adding one more constraint to discrete labeling. Under this stronger condition, we prove that the star graph <img src=image/13441705_07.gif> admits a strong discrete labeling if and only if m is even, highlighting a parity-based constraint absent in standard discrete labeling. These findings expand the known families of graphs with discrete and strong discrete labeling, offering new insights into structural properties that facilitate such labeling. Potential applications include coding theory, network design, and algorithm optimization, while open questions remain regarding the extensibility of these results to other graph operations and more complex graph classes.

A graph G is an ordered pair of sets G(V, E), where V is a set of vertices and the elements of set E are usually called edges. The concept of graph labeling has recently gained a lot of popularity in the area of graph theory. his popularity is due not only to the mathematical challenges of graph labelings but also to the wide range of applications that graph labeling offer to other brances of science, for instance, x-ray, cryptography, coding theory, circuit design and communication network design. A graph labeling we mean an assignment of integers to elements of a graph such as vertex, edge, and both. Local antimagic of a graph was motivated by Arumugam et al.. Thus, we initiate to developed the concept of local antimagic vertex coloring for subdevided graphs. A graph G called sub-devided if the graph G be the graph obtained by inserting a vertex to each edge of the graph G. Definition the concept local antimagic vertex coloring is f : E(G) → {1, 2, 3…, |E(G)|} if for any two adjacent vertices a 1 and a 2, w(a 1) = w(a 2), where for v ∈ G, , where E(a) and V(a) are respectively the set of edges incident to a. The local antimagic vertex labeling induces a proper vertex coloring of graph G if each vertex a is assigned the color w(a). The minimum colors needed to coloring the vertices in graph G called local antimagic vertex chromatic number, denoted by χla (G). In this paper we study the local antimagic vertex coloring of sub-devided some special graph as follows SFn,m , SSn,m , SWn,m and SFn,m .

L(2,1)-colorings and Irreducible No-hole Colorings of Cartesian Product of Graphs