Edge Antimagic Labelling for Security: Utilizing Graph Theory for Improved Security and Quantum-Safe Encryption

A special class of graphs is introduced in this paper. The graphs belonging to this class are characterised by the existence of unique node labels. A number of matching algorithms for graphs with unique node labels are developed. It is shown that problems such as graph isomorphism, subgraph isomorphism, maximum common subgraph (MCS) and graph edit distance (GED) have a computational complexity that is only quadratic in the number of nodes. Moreover, computing the median of a set of graphs is only linear in the cardinality of the set. In a series of experiments, it is demonstrated that the proposed algorithms run very fast in practice. The considered class makes the matching of large graphs, consisting of thousands of nodes, computationally tractable. We also discuss an application of the considered class of graphs and related matching algorithms to the classification and detection of abnormal events in computer networks.

Graph labeling is the assignment of integers to edges or vertices or both depending on some factors which is consequential in the field of graph theory. In this paper, anti-magic graph labeling of two graphs, middle graph and total graph of path graph have been mainly focused. Modified proofs of theorems on anti-magic graph labeling have been introduced here. Both of the theorems are applicable for strong anti-magic graph. The correctness of the proofs has been demonstrated through various examples.

Graph labeling is the process of assigning labels to vertices and edges under certain conditions. This paper investigates the graceful local antimagic labeling of various graph families, excluding symmetric labelings, using computational experiments and Python-based algorithms. Through these experiments, we identify new results and patterns within specific graph classes. The study expands on the existing literature by offering computational evidence, proposing algorithms for the verification of labelings, and exploring the relationship between the local antimagic labeling and the chromatic number. Our results increase the understanding of graph labeling and offer insights into its computational aspects.

The results of the next two sections are based on Somasundaram, Somasundaram (Pattern Recongit Lett 19:787–791, 1998), [29]. The formal mathematical definition of domination was given by Ore, (Theory of graphs. American Mathematical Society, Providence, 1962), [22]. Cockayne and Hedetnieme, (Networks 7:247–261, 1977, [3]), published a survey paper on this topic in 1977 and since then hundreds of papers have been published on this subject. According to Somasundaram, Somasundaram (Pattern Recongit Lett 19:787–791, 1998, [29]), the rapid growth of research in this area is due to the following three factors. (1) The diversity of applications of domination theory to both real world and mathematical coverings or location problems. (2) The wide variety of domination parameters that can be defined. (3) The NP-completeness of the basic domination problem, its close and natural relationship to other NP-complete problems and the subsequent interest in finding polynomial time solutions to domination problems in special classes of graphs.

Antimagic graph labeling is the allotment of positive numbers to the edges of a graph such that the sum of graph edge labels incident to each node varies. The paper emphasizes on strong antimagic graph labeling for splitting graph and shadow graph of cycle. Modified theorems and their proofs on splitting and shadow graphs of cycle have been presented. The validity of the theorems has been illustrated through numerous examples.

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.

The Weisfeiler-Leman procedure is a widely-used approach for graph isomorphism testing that works by iteratively computing an isomorphism-invariant coloring of vertex tuples. Meanwhile, a fundamental tool in structural graph theory, which is often exploited in approaches to tackle the graph isomorphism problem, is the decomposition into bi- and triconnected components. We prove that the 2-dimensional Weisfeiler-Leman algorithm implicitly computes the decomposition of a graph into its triconnected components. Thus, the dimension of the algorithm needed to distinguish two given graphs is at most the dimension required to distinguish the corresponding decompositions into 3-connected components (assuming dimension at least 2). This result implies that for k >= 2, the k-dimensional algorithm distinguishes k-separators, i.e., k-tuples of vertices that separate the graph, from other vertex k-tuples. As a byproduct, we also obtain insights about the connectivity of constituent graphs of association schemes. In an application of the results, we show the new upper bound of k on the Weisfeiler-Leman dimension of graphs of treewidth at most k. Using a construction by Cai, Furer, and Immerman, we also provide a new lower bound that is asymptotically tight up to a factor of 2.

This paper presents a comprehensive formal security framework for key derivation functions (KDF). The major security goal for a KDF is to produce cryptographic keys from a private seed value where the derived cryptographic keys are indistinguishable from random binary strings. We form a framework of five security models for KDFs. This consists of four security models that we propose: Known Public Inputs Attack (KPM, KPS), Adaptive Chosen Context Information Attack (CCM) and Adaptive Chosen Public Inputs Attack(CPM); and another security model, previously defined by Krawczyk [6], which we refer to as Adaptive Chosen Context Information Attack(CCS). These security models are simulated using an indistinguisibility game. In addition we prove the relationships between these five security models and analyse KDFs using the framework (in the random oracle model).

The study examines advanced cybersecurity strategies for health management systems focusing on emerging technologies involving blockchain, artificial intelligence, and the security frameworks of Internet of Medical Things. Health management systems are more increasingly exposed to cybersecurity threats; hence, techniques for the protection of sensitive patient data are strengthened. In this present work, an effort has been made to assess how blockchain can be used effectively for storing data securely and thus achieve a very high penetration of reducing unauthorized access attempts at 97%. AI-driven threat detection systems achieved a 94% strength in the detection of health systems threats while traditional systems were at 12%. Adaptive key rotation for device authentication within IoT security framework resulted in a 92% success rate in guarding IoMT network against a data breach. Therefore, the convergence of these three technologies indicates an all-encompassing multilayered approach to security with enhanced resilience and integrity of data. This area needs constant innovation in light of the security issues to keep up with health management systems that maintain privacy and reliability. Sector-specific adaptations will drive further light on these technologies and provide more robust cybersecurity within the health sector.

Link Prediction (LP) is a fundamental task in graph representation learning, with numerous applications in diverse domains. However, the generalizability of LP models is often compromised due to the presence of noisy or spurious information in graphs and the inherent incompleteness of graph data. To address these challenges, we draw inspiration from the Information Bottleneck principle and propose a novel data augmentation method, COmplete and REduce (CORE) to learn compact and predictive augmentations for LP models. In particular, CORE aims to recover missing edges in graphs while simultaneously removing noise from the graph structures, thereby enhancing the model’s robustness and performance. Extensive experiments on multiple benchmark datasets demonstrate the applicability and superiority of CORE over state-of-the-art methods, showcasing its potential as a leading approach for robust LP in graph representation learning.

This paper presents algorithms , formulas and proofs for antimagic labelings for some graphs which are related in structure with helms. The labelings obtained are not only antimagic but also exhibit much regularity of patterns in the behaviour of the induced vertex labels.

Graph isomorphism completeness for chordal bipartite graphs and strongly chordal graphs

Polar graphs and maximal independent sets

One of the best ways to protect an organization’s assets is to implement security requirements defined by different standards or best practices. However, such an approach is complicated and requires specific skills and knowledge. In case an organization applies multiple security standards, several problems can arise related to overlapping or conflicting security requirements, increased expenses on security requirement implementation, and convenience of security requirement monitoring. To solve these issues, we propose using graph theory techniques. Graphs allow the presentation of security requirements of a standard as graph vertexes and edges between vertexes, and would show the relations between different requirements. A vertex cover algorithm is proposed for minimum security requirement identification, while graph isomorphism is proposed for comparing existing organization controls against a set of minimum requirements identified in the previous step.

The metric dimension of graphs has been extended in some types and variations such as local metric dimension and complement metric dimension of graphs. Merging two concepts is one way of developing the concept of graph theory as a branch of mathematics. These two variations motivated us to construct a new concept of metric dimension so called local complement metric dimension. We apply this new concept to some particular classes of graphs and the corona product of two graphs. We also characterize the local complement metric dimension of graphs with certain properties, namely, bipartite graphs and odd cycle graphs. Furthermore, we discover the local complement metric dimension of the corona product of two particular graphs as well as the corona product of two general graphs.