Search Results (5)

Recent advances in computational chemistry, machine learning, and large-scale virtual screening have rapidly expanded the accessible chemical space, increasing the need for interpretable molecular representations that capture the hierarchical topological structure of molecules. Existing formats, such as Simplified Molecular Input Line Entry System (SMILES) strings and MOL files, effectively encode molecular graphs but provide limited support for representing the multi-level structural information needed for complex downstream tasks. To address these challenges, we introduce HiFrAMes, a novel graph–theoretic hierarchical molecular fragmentation framework that decomposes molecular graphs into chemically meaningful substructures and organizes them into hierarchical scaffold representations. HiFrAMes is implemented as a four-stage pipeline consisting of leaf and ring chain extraction, ring mesh reduction, ring enumeration, and linker detection, which iteratively transforms raw molecular graphs into interpretable abstract objects. The framework decomposes molecules into chains, rings, linkers, and scaffolds while retaining global topological relationships. We apply HiFrAMes to both complex and drug-like molecules to generate molecular fragments and scaffold representations that capture structural motifs at multiple levels of abstraction. The resulting fragments are evaluated using selection criteria established in the fragment-based drug discovery literature and qualitative case studies to demonstrate their suitability for downstream computational tasks. Full article

The theory of semisimple rings plays a fundamental role in noncommutative algebra. We study the complexity of the problem of semisimple rings using the tools of computability theory. Following the general idea of computably enumerable (c.e. for short) universal algebras, we define a c.e. ring as the quotient ring of a computable ring modulo a c.e. congruence relation and view such rings as structures in the language of rings, together with a binary relation. We formalize the problem of being semisimple for a c.e. ring by the corresponding index set and prove that the index set of c.e. semisimple rings is ${\mathsf{\Sigma }}_3^0$-complete. This reveals that the complexity of the definability of c.e. semisimple rings lies exactly in the ${\mathsf{\Sigma }}_3^0$ of the arithmetic hierarchy. As applications of the complexity results on semisimple rings, we also obtain the optimal complexity results on other closely connected classes of rings, such as the small class of finite direct products of fields and the more general class of semiperfect rings. Full article

Let $u,v$, and w be indeterminates over ${\mathbb{F}}_{p^m}$ and let $R={\mathbb{F}}_{p^m}+u{\mathbb{F}}_{p^m}+v{\mathbb{F}}_{p^m}+w{\mathbb{F}}_{p^m},$ where p is a prime. Then, R is a ring of order $p^{4m}$, and $R\cong {\displaystyle \frac{{\mathbb{F}}_{p^m}[u,v,w]}{I}}$ with maximal ideal $J=u{\mathbb{F}}_{p^m}+v{\mathbb{F}}_{p^m}+w{\mathbb{F}}_{p^m}$ of order $p^{3m}$ and a residue field ${\mathbb{F}}_{p^m}$ of order $p^m,$ where I is an appropriate ideal. In this article, the goal is to improve the understanding of linear codes over local non-chain rings. In particular, we investigate the symmetrized weight enumerators and generator matrices of linear codes of length N over $R.$ In order to accomplish that, we first list all such rings up to the isomorphism for different values of the index of nilpotency l of J, $2\le l\le 4.$ Furthermore, we fully describe the lattice of ideals of R and their orders. Next, for linear codes C over $R,$ we compute the generator matrices and symmetrized weight enumerators, as shown by numerical examples. Full article

The Zhang–Zhang polynomial of a benzenoid system is a well-known counting polynomial that was introduced in 1996. It was designed to enumerate Clar covers, which are spanning subgraphs with only hexagons and edges as connected components. In 2018, the generalized Zhang–Zhang polynomial of two variables was defined such that it also takes into account 10-cycles of a benzenoid system. The aim of this paper is to introduce and study a new variation of the Zhang–Zhang polynomial for phenylenes, which are important molecular graphs composed of 6-membered and 4-membered rings. In our case, Clar covers can contain 4-cycles, 6-cycles, 8-cycles, and edges. Since this new polynomial has three variables, we call it the multivariable Zhang–Zhang (MZZ) polynomial. In the main part of the paper, some recursive formulas for calculating the MZZ polynomial from subgraphs of a given phenylene are developed and an algorithm for phenylene chains is deduced. Interestingly, computing the MZZ polynomial of a phenylene chain requires some techniques that are different to those used to calculate the (generalized) Zhang–Zhang polynomial of benzenoid chains. Finally, we prove a result that enables us to find the MZZ polynomial of a phenylene with branched hexagons. Full article

Pentahexagonal annuli are closed chains consisting of regular pentagons and hexagons. Such configurations can be easily recognized in various complex designs, in particular, in molecular carbon constructions. Results of computer enumeration of annuli without overlapping on the plane are presented for up to 18 pentagons and hexagons. We determine how many annuli have certain properties for a fixed number of pentagons. In particular, we consider symmetry, pentagon separation (the least ring-distance between pentagons), uniformity of pentagon distribution, and pentagonal thickness (the size of maximal connected part of pentagons) of annuli. Pictures of all annuli with the number of pentagons and hexagons up to 17 are presented (more than 1300 diagrams). Full article