Article
Mathematics, Applied
Bing-Hau He, Henryk A. Witek
Summary: This study characterizes the number of Clar covers, Kekule structures, and Clar covering polynomials of benzenoid parallelogram chains formed by merging k benzenoid parallelograms. The results, expressed as determinants of structured matrices, provide insight into complex benzenoid moieties in terms of elementary benzenoids.
DISCRETE APPLIED MATHEMATICS
(2021)
Article
Mathematics
Shouliu Wei, Fuliang Lu, Xiaoling Ke
Summary: The article presents the formula for calculating the number of perfect matchings of two types of hexagons on the torus using Pfaffians.
JOURNAL OF MATHEMATICS
(2022)
Article
Mathematics, Applied
Niko Tratnik
Summary: This paper introduces and studies a new variation of the Zhang-Zhang polynomial for phenylenes, which considers different cycles in specific structures. The calculation of this polynomial for phenylenes requires techniques different from those used for benzenoid chains.
Article
Chemistry, Physical
Dong Rong Wu, Peng Yeh, Yueh Ting Chen, Cheng Hao Lu, Jung Yin Hsiao, Elise Y. -T. Li
Summary: In this study, we used triangulene building blocks to design nanographene fragments with different shapes and sizes. We found that these fragments have stable high spin states corresponding to the ferromagnetic arrangement of their constituent triangulene units. By analyzing the structure and bridge geometry, we can modulate the spin multiplicities and exchange energies. These results provide numerical references and valuable empirical rules for the design of all-carbon-based spintronics in the future.
JOURNAL OF PHYSICAL CHEMISTRY C
(2023)
Article
Chemistry, Physical
Jack E. Graver, Elizabeth J. Hartung, Aaron Williams
Summary: This paper investigates the influence of different chiral indices on the conjugated pi-systems in capped nanotubes, revealing a fully conjugated pi-system along the nanotube's cylinder when the chiral index n-m is a multiple of 3, while fracture lines break the pattern if it is not a multiple of 3. These results are closely related to whether the nanotube is metallic or semiconductor.
Article
Mathematics, Applied
Boris Furtula, Slavko Radenkovic, Izudin Redzepovic, Niko Tratnik, Petra Zigert Pletersek
Summary: This paper introduces the introduction of the generalized Zhang-Zhang (GZZ) polynomial and its application in increasing the sensitivity to pi-electron cyclic conjugation of 10-membered rings. The recursive formulas for calculating the GZZ of benzenoid systems are derived, and an algorithm for calculating the GZZ of benzenoid chains is provided. Lastly, the chemical applicability of GZZ is tested.
APPLIED MATHEMATICS AND COMPUTATION
(2022)
Article
Chemistry, Multidisciplinary
Henryk A. Witek, Rafal Podeszwa, Johanna Langner
Summary: A closed-form formula for the ZZ polynomials of hexagonal graphene flakes with specific structural parameters is reported, expanding the available information by a factor of 10. The main purpose of presenting these numerical results is to provide reference data for the chemical and mathematical communities to derive, understand, and test general ZZ polynomial formulas for hexagonal flakes with arbitrary parameters.
MATCH-COMMUNICATIONS IN MATHEMATICAL AND IN COMPUTER CHEMISTRY
(2021)
Article
Chemistry, Physical
Yang Wang
Summary: In this study, a quantitative theory of chemical resonance based on Clar-type resonance structures (Clar resonators) was developed and applied to a large-scale resonance analysis of PAHs. By constructing wave functions of Clar resonators and calculating their weights and one-electron energies, the general validity of the Clar rule was confirmed. Based on this analysis, three extended Clar rules and a unified quantitative model were proposed.
JOURNAL OF PHYSICAL CHEMISTRY A
(2022)
Article
Chemistry, Multidisciplinary
Yutong Liu, Congcong Ma, Haiyuan Yao, Xu Wang
Summary: The paper proposes an efficient method - integer linear programming, to compute the forcing number and anti-forcing number of a given perfect matching in a graph. As applications, the di-forcing polynomials of C-60, C-70, and C-72 are obtained, along with their forcing and anti-forcing polynomials.
MATCH-COMMUNICATIONS IN MATHEMATICAL AND IN COMPUTER CHEMISTRY
(2022)
Article
Chemistry, Multidisciplinary
Johanna Langner, Henryk A. Witek
Summary: The algorithm presented in this paper calculates the ZZ polynomial of regular strips through four steps, including constructing the corresponding poset, constructing the set of linear extensions, computing the number of descents, and computing the number of fixed labels. The practical applications of the algorithm are demonstrated with several examples, showcasing the computation of ZZ polynomials for regular m-tier benzenoid strips.
MATCH-COMMUNICATIONS IN MATHEMATICAL AND IN COMPUTER CHEMISTRY
(2022)
Article
Chemistry, Multidisciplinary
Juan Andres Montoya
Summary: This study investigates the algorithmic hardness of counting problems in crystal physics and fullerene chemistry, claiming them to be representative of mathematical nanosciences and observing their sparsity. By analyzing the complexity class #P-1, it is concluded that these seemingly hard problems cannot be hard for NP. Additionally, conjectures and weak results related to counting matchings, Hamiltonian cycles, and Clar sets are discussed in the paper.
MATCH-COMMUNICATIONS IN MATHEMATICAL AND IN COMPUTER CHEMISTRY
(2021)
Article
Chemistry, Organic
Shuang Zhao, Xinman Fan
Summary: This paper explores the relationship between the Clar number and forcing number in hexagonal systems and introduces the concept of forcing polynomials. By deriving a recurrence relation for the forcing polynomial of double hexagonal chains, explicit forms of forcing polynomials for double linear and zigzag hexagonal chains, as well as some special examples, are obtained.
POLYCYCLIC AROMATIC COMPOUNDS
(2023)
Article
Chemistry, Medicinal
Yang Wang
Summary: The Fries rule, a simple and intuitive tool for predicting dominant Kekule structures of PAHs, has been verified, generalized, and quantified based on resonance analysis of over 1500 PAH and fullerene molecules. The extended rules, counting the numbers of electrons within all rings, can rank the relative importance of Kekule structures for all considered systems. The study also proposes a graph-based aromaticity indicator applicable to PAHs and fullerenes, which does not require any quantum chemistry calculations and can predict molecular properties related to local aromaticity.
JOURNAL OF CHEMICAL INFORMATION AND MODELING
(2022)
Article
Multidisciplinary Sciences
Ali N. A. Koam, Ali Ahmad, M. A. Asim, Muhammad Azeem
Summary: In chemistry, the structure of chemical compounds is commonly represented using graphs. Vertices or nodes represent atoms, and edges represent bond types. The resolving set of mixed metric dimension, known as F-m, refers to a subset of vertices in a graph where each node and edge has a distinct representation or location. This concept helps determine the unique position of a structure or graph and finds application in studying drug patterns in pharmaceutical research.
JOURNAL OF KING SAUD UNIVERSITY SCIENCE
(2022)
Article
Mathematics, Applied
Maryam Salem Alatawi, Ali Ahmad, Ali N. A. Koam, Sadia Husain, Muhammad Azeem
Summary: In this paper, the exact metric and fault-tolerant metric dimension of the benzenoid tripod structure are determined. The generalized version of this parameter is also computed and it is proved that all the parameters are constant. Resolving sets provide a unique representation for chemical structures and have important applications in fields such as pharmaceutical research.
