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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 R0 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.

Graphs in the field of science and technology make considerable use of theoretical concepts. When dealing with numerous links and circumstances in which there are varying degrees of ambiguity or robustness in the connections between aspects, rather than purely binary interactions, cubic fuzzy graphs (CFGs) are more adaptable and compatible than fuzzy graphs. To better represent the complexity of interactions or linkages in the real world, an emerging CFG can be very helpful in achieving better problem-solving abilities that specialize in domains like network analysis, the social sciences, information retrieval, and decision support systems. This idea can be used for a variety of uncertainty-related issues and assist decision-makers in selecting the best course of action through the use of a CFG. Enhancing the maximized network of three cubic fuzzy graphs’ decision-making efficiency was the ultimate objective of this study. We introduced the maximal product of three cubic fuzzy graphs to investigate how interval-valued fuzzy membership, fuzzy membership, and the miscellany of relations are all simultaneously supported through the aspect of degree and total degree of a vertex. Furthermore, the domination on the maximal product of three CFGs was illustrated to analyze the minimum domination number of the weighted CFG, and the proposed approach is illustrated with applications.

Domination plays an indispensable role in graph theory. Various types of domination explore various types of applications. Equal-status people work together and interlace with each other easily. In this paper, the paired equitable domination of a graph, its inflated graph, and its complement of an inflated graph were studied. The relationship between the domination number of the graph, the equitable domination number, and the paired equitable domination number of complements of the inflated graph were established. Furthermore, we proved the Nordhaus–Gaddum-type inequality, that is, γpre(H)+γpre(H)≤6 if H is a graph with m nodes where m≡0, 2(mod 8) and d(ai) = (m/2) for all ai. The challenges and limitations of this parameter of paired equitable and equitable domination depends on the degree of the vertex of the graph. Practical applications are discussed in various fields and illustrated using the studied parameter.

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.

An optimal network refers to a computer or communication network designed, configured, and managed to maximize efficiency, performance, and effectiveness while minimizing cost and resource utilization. In a network design and management context, optimal typically implies achieving the best possible outcomes between various factors. This research investigated the use of fuzzy graph edge coloring for various fuzzy graph operations, and it focused on the efficacy and efficiency of the fuzzy network product using the minimal spanning tree and the chromatic index of the fuzzy network product. As a network made of nodes and vertices, measurement with vertices is a parameter for domination, and edge measurement is a parameter for edge coloring, so we used these two parameters in the algorithm. This paper aims to identify an optimal network that can be established using product outcomes. This study shows a way to find an optimal fuzzy network based on comparative optimal parameter domination and edge coloring, which can be elaborated with applications. An algorithm was generated using an optimal approach, which was subsequently implemented in the form of applications.

Let Heff (Veff, Eeff) be a finite simple connected graph of order m with vertex set Veff and edge set Eeff. A dominating set Sds⊆Veff is called an efficiently dominating set if, for every vertex ua∈VG, NGua∩Sds=1—where NG [ua] denotes the closed neighborhood of the vertex ua. 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.