Journal of Computational Mathematica

Advancing Post-Cryptographic Schemes Using Number Theoretic Structures: Algorithms, Security, and Real World Implementation

2026-07-03T07:41:32+00:00

Cryptography, a term rooted in Greek that translates to 'secure writing,' encompasses the intricate study of methods employed to encrypt and decrypt information, ensuring secure communication between parties. This fundamental discipline plays a critical role in safeguarding sensitive data from unauthorized access and malicious entities. The process begins with encryption, where plain text, or the original readable information, is transformed into ciphertext through specific algorithms and protocols. This transformation obscures the content, making it unintelligible to anyone who does not possess the appropriate keys or credentials to decode it. Decryption is the complementary process that reverses this transformation, restoring the ciphertext to its original plain text format. This allows authorized users to access the information securely while preventing would-be intruders from deciphering its contents. In addition to these basic principles, modern cryptography incorporates various techniques, such as symmetric and asymmetric encryption, digital signatures, and hash functions, each serving specific purposes in the realm of data protection and secure communication. Overall, cryptography is essential in today's digital landscape, underpinning everything from online banking transactions to secure messaging systems and helping to ensure the integrity and privacy of information exchanged over insecure networks.

Super Mean Graph Labeling: A Novel Cryptographic Framework Using Five-Star Graphs with Applications in Secure Communications and Epidemic Modeling

2026-07-03T07:52:22+00:00

This paper introduces an innovative cryptographic framework based on super mean labeling of five-star graphs K1,V1 ∪ K1,V2 ∪ K1,V3 ∪ K1,V4 ∪ K1,V5, where V1 ≤ V2 ≤ V3 ≤ V4 ≤ V5. We present a comprehensive mathematical foundation for super mean labeling and develop systematic methodologies for encoding messages through carefully constructed graph structures. Three distinct implementation approaches are demonstrated, incorporating computational techniques through C programming for alphabetical mapping, including subtraction-based and division-based numbering schemes. Each approach is thoroughly illustrated with complete message encoding examples, visual cryptography representations, and detailed security analysis. Beyond cryptographic applications, we extend the framework to epidemic modeling, demonstrating how super mean labeling can represent complex disease transmission dynamics in multi-population systems. The epidemic modeling application includes complete mathematical formulations of multi-population SIR models, transmission matrix encoding through graph labeling, and computational implementations for disease surveillance. Our results establish that five-star graphs with super mean labeling provide optimal balance between encoding capacity and structural complexity, offering robust security through multiple layers of mathematical obfuscation. The integration of graph theory with computational algorithms creates a versatile framework applicable to both secure communications and public health informatics.

Detailed Proposed Algorithm: Symmetric Encryption and Decryption Using Eulerian Circuits in Simple Graphs

2026-07-10T08:03:54+00:00

This paper proposes a novel symmetric-key encryption scheme using simple weighted graphs and Eulerian circuits. Unlike NP-complete Hamiltonian cycle methods, it employs polynomial-time Eulerian circuits via Hierholzer‘s algorithm. Plaintext is mapped to a weighted graph using a secret key. An Eulerian circuit generates a dynamic key and Euler Tour Matrix. Encryption uses matrix multiplications with a shared upper triangular key and modular reduction. Decryption reverses the operations using matrix inverses. Statistical tests confirm strong diffusion and randomness. The scheme provides better efficiency and scalability than traditional Hamiltonian-based approaches.

Graph-Based Cryptographic Encoding and Epidemiological Modeling via Even Felicitous Labeling

2026-07-10T08:22:17+00:00

In this work, we have identifified a way for encoding a secret message utilizing the GMJ (Graph Message Jumbled) Code and even felicitous labeling on fifive-star graphs K1,σ1 ∪ K1,σ2 ∪ K1,σ3 ∪ K1,σ4 ∪ K1,σ5 . For every star graph, we give two examples and use a Java program to apply alphabetical techniques to the Corona Number and Triangular Number. A strategy for labeling an even felicitous graph and flflowchart is provided for transforming plaintext into ciphertext (Picture Coding). Furthermore, we demonstrate an innovative application of these graph labeling techniques to epidemiological modeling through a detailed analysis of the SIR (Susceptible-Infected-Recovered) epidemic model on star networks. The paper shows how felicitous labeling can encode varying transmission probabilities and infection states, providing insights for targeted public health interventions. This interdisciplinary approach bridges cryptography, graph theory, and epidemiology, offffering both secure communication methods and practical disease spread modeling frameworks.

A Study on Vertex Coloring and Edge Coloring and its Application

2026-08-10T04:50:21+00:00

Graph coloring is an important area of graph theory that deals with assigning colors to the vertices and edges of a graph under certain conditions. Vertex coloring ensures that adjacent vertices have different colors, while edge coloring ensures that adjacent edges receive different colors. This study presents the basic concepts of vertex coloring and edge coloring and discusses their practical applications in scheduling, timetabling, frequency assignment, map coloring, and resource allocation. These coloring techniques help in minimizing conflicts and optimizing the efficient use of resources in various real-world problems.

A Mathematical Framework for Social Network Analysis Using Graph Theory

2026-08-10T04:56:17+00:00

The rapid growth of online social networking platforms such as Facebook, Twitter, and WhatsApp has created complex interaction structures that cannot be fully understood using traditional statistical methods alone. This study aims to model and interpret the structural properties of social networks using graph theory as a mathematical framework. In this representation, users are modeled as vertices and interactions such as friendships, follows, and message exchanges are represented as edges. The methodology involves constructing directed and undirected graphs from real-world inspired data and analyzing them using graph-theoretic concepts such as degree distribution, connectivity, paths, cycles, bridges, cut-vertices, bipartite graphs, and Eulerian properties. Important theorems including Eu- ler’s theorem and connectivity principles are applied to identify influential users and com- munication bottlenecks. The results show that the platforms exhibit struc- tures: WhatsApp networks are highly clustered and nearly connected. The study demonstrates that graph theory provides an ef- fective tool for understanding information flow, network stability, and influence patterns in social networks.

A Discrete-Time Compartmental Model of Problematic Pornography Use Dynamics with Clinical Intervention, Analysed Using Discrete Physics-Informed Neural Networks (D-PINNs)

2026-08-10T05:03:15+00:00

Continuous-time compartmental models, formulated as systems of ordinary differential equations, are the dominant mathematical framework for describing the population-level
dynamics of behavioural-health conditions such as problematic pornography use. In practice, however, the longitudinal data available to clinicians and researchers, periodic clinical-screening waves, survey administrations, or treatment-program intake records, are inherently discrete, sampled at fixed reporting intervals rather than continuously. This paper develops and analyses a discrete-time counterpart to a continuous compartmental model of problematic pornography use dynamics under clinical intervention, formulated directly as a system of nonlinear difference equations rather than as a discretisation applied after the fact. The model partitions a population into four interacting groups, evaluated at discrete reporting indices n = 0,1,2,...: susceptible individuals (Sn), individuals exhibiting problematic or compulsive use patterns (Pn), individuals engaged in clinical treatment (Cn), and recovered individuals (Rn). We establish the positivity and boundedness of solutions, locate the problematic-use-free and persistent-use equilibria, which coincide algebraically with those of the underlying continuous system, and derive the same basic reproduction number R0 via the Next Generation Matrix method. Local stability is then analysed using the discrete-time Jacobian and the Jury (discrete Routh–Hurwitz) stability criterion, requiring all eigenvalues to lie strictly within the unit circle, a materially different and generally more restrictive condition than the continuous requirement of negative real part. We show analytically that the discrete and continuous eigenvalues are related by a unit shift, and derive an explicit, computable sufficient condition under which continuous-time stability at R0 < 1 carries over to the discrete-time system. To complement this analysis, we implement a discrete Physics-Informed Neural Network (D-PINN) that enforces the governing difference equations directly as a residual loss, without requiring automatic differentiation, and use it to solve the forward and inverse problems from synthetic discretely sampled observational data. The trained network reproduces the forward trajectories with normalized root-mean-square errors below 0.55 across all four compartments, and recovers three of the four unknown parameters with relative error under 10%, while the clinical-intervention-efficacy parameter is markedly more weakly identifiable under discrete, sparser sampling than in the continuous-time formulation, a finding we discuss in detail.

Mathematical Modeling of Internet Gaming Disorder Dynamics with Behavioral Intervention, Using Physics-Informed Neural Networks (PINNs)

2026-08-10T05:15:06+00:00

Internet Gaming Disorder, formally recognised as a distinct condition in the World Health Organization’s eleventh revision of the International Classification of Diseases (ICD-11), has emerged as a growing behavioural-health concern, yet quantitative, dynamic frameworks for describing how such patterns emerge within a population and respond to behavioral intervention over time remain scarce relative to the extensive cross-sectional and clinical-assessment literature on the topic. In this work we develop and analyse an original compartmental mathematical model for the dynamics of Internet Gaming Disorder under active behavioral intervention. The model partitions a population into four interacting groups: susceptible individuals (S), individuals exhibiting Internet Gaming Disorder (G), individuals engaged in behavioral or clinical intervention (I), and recovered individuals who have re-established regulated gaming habits and exert a protective effect on their peers (R). The interactions among these groups are governed by a system of four coupled nonlinear ordinary differential equations. We establish the positivity and boundedness of solutions, identify a disorder-free equilibrium and a persistent-disorder equilibrium, and derive the basic reproduction number R0 using the Next Generation Matrix method. Local stability of both equilibria is analysed via the Jacobian matrix, characteristic polynomial, and Routh–Hurwitz criteria, showing that the disorder-free state is locally asymptotically stable whenever R0 < 1. To complement this analytical treatment, we implement a Physics-Informed Neural Network (PINN) that embeds the governing differential equations into its training objective, and use it to (i) solve the forward problem of reconstructing the population trajectories and (ii) solve the inverse problem of recovering unknown behavioural parameters directly from simulated observational data. The trained network reproduces the forward trajectories with normalized root-mean-square errors below 0.4, and recovers three of the four unknown parameters with relative error under 11%, while the weakly identifiable intervention-efficacy parameter is recovered with a larger, but bounded, error, a finding we discuss in terms of parameter identifiability. This hybrid analytical-computational framework offers a reproducible, quantitative basis for studying, and ultimately informing, behavioral and clinical intervention strategies aimed at Internet Gaming Disorder.

Fixed Point Results for Generalized Rational Type (ϑ, ψ,ϕ)-Weak Contractive Mappings in b-Metric Spaces with Application

2026-08-14T06:11:00+00:00

This problem concentrates on existence and uniqueness of common fixed point for two pairs of self mappings gratify broad rational type (ϑ,ψ,ϕ)- weak contractive axiom in the framework of b-Metric Space. The pairs of mappings are weakly compatible and one pair is α- admissible w.r.t. the other. Our results broad and improve some well known results in the literature. As an application of our result, we shall discuss the existence of common solution of integral equations.

Super Metric Spaces and Fixed Point Results using (E.A) and (CLRP) Properties

2026-08-14T08:00:51+00:00

In this article, we established the common fixed point results by using property (E.A) and (CLRP) property in the framework of Super metric spaces. The obtained results extend several well-defined findings in classical metric spaces to the more general setting of Super metric spaces. Furthermore, suitable example is provided to demonstrate the effectiveness and applicability of the proposed results.