The growing sophistication of cyber-attacks demands encryption processes that go beyond the confines of conventional cryptographic methods. Traditional cryptographic systems based on numerical algorithms or standard graph theory are still open to structural and computational attacks, particularly in light of advances in computation power. Fuzzy logic’s in-built ability to manage uncertainty together with the representation ability of fuzzy hypergraphs for describing complex interrelations offers an exciting avenue in the direction of developing highly evolved and secure cryptosystems. This paper lays out a new framework for cryptography using fuzzy hypergraph networks in which a hidden value is converted into a complex structure of dual fuzzy hypergraphs that remains completely connected. This technique not only increases the complexity of the encryption process, but also significantly enhances security, thus making it highly resistant to modern-day cryptographic attacks and appropriate for high security application. This approach improves security through enhanced entropy and the introduction of intricate multi-path data exchange through simulated nodes, rendering it highly resistant to contemporary cryptographic attacks. It ensures effective key distribution, accelerated encryption–decryption processes, and enhanced fault tolerance through dynamic path switching and redundancy. The adaptability of the framework to high-security, large-scale applications further enhances its robustness and performance.

Cryptography is a pivotal application of graph theory in ensuring secure communication systems. Modern cryptography is deeply rooted in mathematical theory and computer science practices. It is widely recognized that encryption and decryption processes are primarily outcomes of mathematical research. Given the increasing importance of safeguarding secret information or messages from potential intruders, it is imperative to develop effective technical tools for this purpose. These intruders are often well-versed in the latest technological advancements that could breach security. To address this, our study focuses on the efficacious combinatorial technique of graph networks using efficient domination and total magic labeling. The introduction of a graph network based on total magic labeling can significantly influence the network’s performance. This research introduces a novel combinatorial method for encrypting and decrypting confidential numbers by leveraging an efficient dominant notion and labeled graph.

Let H_eff (V_eff, E_eff) be a finite simple connected graph of order m with vertex set V_eff and edge set E_eff. A dominating set $S_{ds}\subseteq V_{eff}$ is called an efficiently dominating set if, for every vertex $u_a\in V_G,\hspace{0.33em}\left|N_G\left[u_a\right]\cap S_{ds}\right|=1$—where NG [u_a] denotes the closed neighborhood of the vertex u_a. Using efficient domination techniques and labelling, we constructed the fuzzy network. An algorithm has been framed to encrypt and decrypt the secret information present in the network, and furthermore, the algorithm has been given in pseudocode. The mathematical modelling of a strong fuzzy network is defined and constructed to elude the burgeoning intruder. Using the study of the efficient domination of fuzzy graphs, this domination parameter plays a nuanced role in encrypting and decrypting the framed network. One of the main purposes of fuzzy networks is encryption, so one of our contributions to this research is to build a novel combinatorial technique to encrypt and decode the built-in fuzzy network with a secret number utilizing effective domination. An illustration with an appropriate secret message is provided along with the encryption and decryption algorithms. Furthermore, we continued this study in intuitionistic fuzzy networks.

Around the world, the people are simultaneously susceptible to or infected with several infections. This work aims at the analysis of the dynamics of transmission of two deadly viruses, COVID-19 and Influenza, using a co-infection epidemiological model by applying the Caputo fractional derivative. Fractional differential equations are currently used worldwide to model physical and biological phenomena. Our comprehension of complicated phenomena is improved when fractional-order derivatives are used to model systems with memory effects and long-range interactions. Mathematical depictions of infectious disease dynamics and dissemination across communities are provided by epidemiological models, which are essential resources for understanding and controlling infectious diseases. These models support informed decision making to prevent outbreaks, evaluate intervention measures, and help researchers and policymakers understand how diseases spread. A subclass of epidemiological models called co-infection models focuses on studying the dynamics of several infectious illnesses that occur in the same population at the same time. They are especially useful in situations where people are simultaneously susceptible to or infected with several infections. Co-infection models provide information on the development of effective control techniques, the progression of disease, and the interactions between several pathogens. The qualitative study via stability analysis is discussed at equilibrium, the reproduction number $R_0$ is computed, and the results are simulated using the Laplace Adomian Decomposition Method (LADM) for Fractional Differential Equations. We employ MATLAB R2023a for graphical presentations and numerical simulations.

This research investigates the integration of neutrosophic logic with Graph Neural Networks (GNNs) for disease transmission risk prediction. Neutrosophic Graph Neural Networks (NGNNs) extend the capabilities of Fuzzy Graph Neural Networks (F-GNNs) as utilized in this paper to process structured information. By merging neutrosophic logic with GNNs, the technique significantly improved the handling of uncertainty while monitoring contagious diseases. The standard GNN model fails to process unreliable and unspecific information commonly found in epidemiological data collections. The neutrosophic triplets \((\mathcal {T, I, F})\) within NGNNs represent uncertain components found in disease outbreak conditions. Improved message passage through the NGNN works best while processing complex and extended datasets that programmers structure with Python. The risk evaluation and transmission uncertainty of diseases appear in visual heat maps to assist in the analysis of network dynamics. Through visual output, the health sector receives essential information related to outbreak development for making better decisions. By uniting state-of-the-art computing solutions with health-related implementation, this approach establishes superior standards for medical risk evaluation and produces foundational capabilities for future research. The authors established a firm foundation for advanced epidemic simulation and risk assessment through their work, which integrates theoretical graph theory advances into practical public health scenarios.

Intuitionistic fuzzy graphs (IFGs) are a powerful tool for modeling uncertainty and complex relationships. They offer versatile frameworks for addressing real‐world challenges. In this research, we have introduced intuitionistic fuzzy vertex order coloring (IFVOC) and analyzed the alpha‐strong (), beta‐strong (), and gamma‐strong () vertices through their degree. We explored important theorems based on the types of strong vertices, broadening the scope of our study. We analyzed multiple IFG products to determine the most optimal network based on some important metrics, including the weight and total number of vertices, the chromatic number, and the weight of the graph’s minimum spanning tree. We systematically evaluate different types of IFG products, such as conormal, modular, residue, and maximal, and consider their implications for fuzzy graph structure and connectivity. We have investigated new metrics to measure the presence and importance of vertices in the graph, assessing both their frequency and overall impact. Additionally, we have investigated the impact of product operations on the significance of nodes and the resultant minimum spanning tree, providing insights into the overall robustness and efficiency of each product. We have explored how variations in product operations impact the distribution of vertices and the characteristics of the minimum spanning tree and chromatic number, elucidating the trade‐offs between different product options. We have modeled the product graph as a network to represent complex systems composed of interconnected entities. The outcomes of this research contribute to the advancement of IFG theory and its use in various fields, such as network design, strategic planning, and system analysis.