This analysis emphasizes the significance of radiation and chemical reaction effects on the boundary layer flow (BLF) of Casson liquid over a linearly elongating surface, as well as the properties of momentum, entropy production, species, and thermal dispersion. The mass diffusion coefficient and temperature-dependent models of thermal conductivity and species are used to provide thermal transportation. Nonlinear partial differential equations (NPDEs) that go against the conservation laws of mass, momentum, heat, and species transportation are the form arising problems take on. A set of coupled dimensionless partial differential equations (PDEs) are obtained from a set of convective differential equations by applying the proper non-similar transformations. Local non-similarity approaches provide an analytical approximation of the dimensionless non-similar system up to two degrees of truncations. The built-in Matlab (Version: 7.10.0.499 (R2010a)) solver bvp4c is used to perform numerical simulations of the local non-similar (LNS) truncations.

This paper presents three novel encryption and decryption schemes based on graph theory that aim to improve security and error resistance in communication networks. The novelty of this work lies in the application of complete bipartite graphs in two of the schemes and the Cartesian product of graphs in the third, representing a unique approach to cryptographic algorithm development. Unlike traditional cryptographic methods, these graph-based schemes use structural properties of graphs to achieve robust encryption, providing greater resistance to attacks and corruption. Each scheme is illustrated with detailed examples that show how the algorithms can be successfully implemented. The algorithms are written in standard mathematical notation, making them adaptable for machine implementation and scalable for real-world use. The schemes are also rigorously analyzed and compared in terms of their temporal and spatial complexities, using Big $\mathcal{O}$ notation. This comprehensive evaluation focuses on their effectiveness, providing valuable insights into their potential for secure communication in modern networks.

Access to clean water is a pressing global concern and membrane technologies play a vital role in addressing this challenge. Thin-film composite membranes prepared via interfacial polymerization (IPol) using meta-phenylenediamine (MPD) and trimesoyl chloride (TMC) exhibit excellent separation performance, but face limitations such as fouling and low hydrophilicity. This study investigated the interaction between MPD and sulfonated zinc phthalocyanine, Zn(SO₂⁻)₄Pc, as a potential strategy for enhancing membrane properties. Using Density Functional Theory (DFT) and Time-Dependent DFT (TD-DFT), we analyzed the optimized geometries, electronic structures, UV–Vis absorption spectra, FT-IR vibrational spectra, and molecular electrostatic potentials of MPD, Zn(SO₂⁻)₄Pc, and their complexes. The results show that MPD/Zn(SO₂⁻)₄Pc exhibits reduced HOMO-LUMO energy gaps and enhanced charge delocalization, particularly in aqueous environments, indicating improved stability and reactivity. Spectroscopic features confirmed strong interactions via hydrogen bonding and π–π stacking, suggesting that Zn(SO₂⁻)₄Pc can act as a co-monomer or additive during IPol to improve polyamide membrane functionality. A conformational analysis of MPD/Zn(SO₂⁻)₄Pc was conducted using density functional theory (DFT) to evaluate the impact of dihedral rotation on molecular stability. The 120° conformation was identified as the most stable, due to favorable π–π interactions and intramolecular hydrogen bonding. These findings offer computational evidence for the design of high-performance membranes with enhanced antifouling, selectivity, and structural integrity for sustainable water treatment applications.

Circulant networks are a very important and widely studied class of graphs due to their interesting and diverse applications in networking, facility location problems, and their symmetric properties. The structure of the graph ensures that it is symmetric about any line that cuts the graph into two equal parts. Due to this symmetric behavior, the resolvability of these graph becomes interning. Subdividing an edge means inserting a new vertex on the edge that divides it into two edges. The subdivision graph G is a graph formed by a series of edge subdivisions. In a graph, a resolving set is a set that uniquely identifies each vertex of the graph by its distance from the other vertices. A metric basis is a resolving set of minimum cardinality, and the number of elements in the metric basis is referred to as the metric dimension. This paper determines the minimum resolving set for the graphs $H_l[1,k]$ constructed from the circulant graph $C_l[1,k]$ by subdividing its edges. We also proved that, for $k=2,3$, this graph class has a constant metric dimension.

In this article, we propose a novel integral transform coined as quaternion quadratic phase S-transform (Q-QPST), which is an extension of the quadratic phase S-transform and study the uncertainty principles associated with the Q-QPST. The Q-QPST possesses some desirable characteristics that are absent in conventional time-frequency transforms, especially for dealing with the time-varying quaternion-valued signals. First, we propose the definition of Q-QPST and then we explore some mathematical properties of the of quaternion Q-QPST, including the linearity, modulation, shift, orthogonality relation, and reconstruction formula. Second, we derive the associated Heisenberg’s uncertainty inequality and the corresponding logarithmic version for Q-QPST. Finally, an illustrative example and some potential applications of the Q-QPST are introduced.