Topological indices (TIs) are numerical tools widely applied in chemometrics, biomedicine, and bioinformatics for predicting diverse physicochemical attributes and biological activities within molecular structures. Despite their significance, the challenges in deriving TIs necessitate novel approaches. This study addresses the limitations of conventional methods in dealing with dynamic molecular structures, focusing on the neighborhood M-polynomial (NM-polynomial), a pivotal polynomial for calculating degree-based TIs. Current literature acknowledges these polynomials but overlooks their limited adaptability to intricate biopolymer relationships. Our research advances by computing degree-based and neighborhood degree-based indices for prominent biopolymers, including polysaccharides, poly--glutamic acid, and poly-L-lysine. Through innovative utilization of the NM-polynomial and the M-polynomial, we establish a fresh perspective on molecular structure and topological indices. Moreover, we present diverse graph representations highlighting the nuanced correlations between indices and structural parameters. By systematically investigating these indices and their underlying polynomials, our work contributes to predictive modelling in various fields. This exploration sheds light on intricate biochemical systems, offering insights into applications encompassing medicine, the food industry, and wastewater treatment. This research deepens our understanding of complex molecular interactions and paves the way for enhanced applications in diverse industries.

The progress made in the field of quantum computing has drawn the attention of the cryptographic community to the existing systems’ security. Thus, researchers are looking for post-quantum cryptographic solutions that would be able to defend against the upcoming quantum attacks. To this end, different cryptographic approaches have been proposed. Among these approaches, lattice-based cryptography stands out as a promising candidate because of it’s efficient parameters and performance characteristics. The robustness of these post-quantum methods has been tested against the NIST criteria so that the security they offer can be trusted. In this work, we conducted a systematic literature review on lattice-based KEM and signature schemes. We have analyzed 41 papers taken from five major databases and addressed four research questions that include the current landscape of lattice-based post-quantum cryptography, the gaps in the existing literature on lattice-based post-quantum cryptography, identifying mathematical hard problems used in the papers, and future research directions. The reviewed papers primarily focus on key encapsulation mechanisms (KEM) and signature schemes. One of the key findings of this study is the identification of promising lattice-based cryptographic schemes beyond the NIST finalists, such as Qu-Octonion NTRU, Tyber, and HAETAE etc., which clearly shows there is notable progress in lattice-based post-quantum systems. In addition, this review identifies key gaps in current research, including a lack of emphasis on the tradeoffs between security and efficiency, and an insufficient design of hybrid cryptosystems. Finally, the strengths and limitations of the current lattice-based Cryptosystems and future research directions are presented.

This paper studies a special case of group divisible designs (GDDs) called 3-GDDs, which were defined by extending the definitions of a group divisible designs and a t-design. In particular, the paper looks at a 3-GDD(n, 4, 5; μ1, μ2) with 4 groups and block size 5. Necessary conditions for the existence given.

A -GDD (, , , , ) was defined by combining the definitions of a group divisible design and a t-design. In this paper, we extend the definitions to 3 groups and block size 5, and we denote such GDD by -GDD (, , , , ). Some necessary conditions for the existence of these GDDs are developed, and several new constructions and specific instances of nonexistence are given.