Chemical Graph Theory Research Articles

σ Interference: Through-Space and Through-Bond Dichotomy.

Dividing orbital interactions into through-space (TS) and through-bond (TB) modes is valuable for understanding various molecular properties. In this paper, we elucidate how the quantum interference phenomenon known as σ interference in electron transport through σ systems arises from TS and TB interactions. We performed electron transport calculations using a combination of density functional theory and nonequilibrium Green's function methods, focusing on ethylenediamine, a classical molecule that effectively highlights the contrast between TS and TB interactions. Our results confirm that destructive σ interference occurs in the syn and gauche conformers of this molecule. To further investigate both TS and TB interactions, we employed two analytical methods: the fragment molecular orbital (FMO) method, which captures the effects of both TS and TB interactions, and the chemical graph theory method, which specializes in TB interactions. The FMO analysis demonstrated that TB interactions lead to the characteristic distribution and energy level alignment of the frontier orbitals. Additionally, it was clarified that a change in TS interaction, due to a variation in the dihedral angle of the molecule, alters the energy gap between these orbitals, resulting in the manifestation of σ interference in the syn and gauche conformers, but not in the trans conformer. The chemical graph theory analysis based on the ladder C model, aimed at exploring the topological origin of σ interference from the network of TB interactions, revealed that σ interference is caused by the cancellation between the walk associated with geminal interactions (σ-conjugation) and the one related to vicinal interaction (σ-hyperconjugation). Notably, it was found that the vicinal interaction, which changes sign with the dihedral angle, has a decisive influence on whether this cancellation occurs. These findings clarify that σ interference arises from the interplay between TS and TB interactions. This insight will be valuable for designing molecular systems that utilize σ interference.

Counting Polynomials in Chemistry II

Some polynomials find their way into chemical graph theory less often than others. They could provide new ways of understanding the origins of regularities in the chemistry of specific classes of compounds. This study’s objective is to depict the place of polynomials in chemical graph theory. Different approaches and notations are explained and levelled. The mathematical aspects of a series of such polynomials are put into the context of recent research. The directions in which this project was intended to proceed and where it stands right now are presented.

A new spectral-based index for graphs and its application to polyaromatic compounds

In this study, we introduce a novel index called the modified inverse sum indeg (MISI) energy as an extension of the traditional inverse sum indeg (ISI) energy and neighbor degree sum energy. We devised an algorithm that employs the simplified molecular input line entry system (SMILES) formula of compounds to compute the MISI energy of polycyclic aromatic hydrocarbon (PAH). Additionally, we established the bounds of the MISI energy for certain classes of graphs with fixed number of vertices. A strong correlation between the MISI energy and the total [Formula: see text]-electron energy of PAHs is observed through regression analysis. Remarkably, this correlation outperforms that achieved by the classical ISI energy. These findings highlight the enhanced effectiveness of the MISI energy as a descriptor for capturing significant molecular properties. Moreover, our study establishes a foundation for further theoretical extensions and practical applications in the field of chemical graph theory.

Computational analysis of diameter eccentricity based and hyper diameter eccentricity based indices for linear saturated monocarboxylic acids

Chemical graph theory deals with chemical graphs, which are used to represent chemical systems. Topological indices of molecular graphs are an important area of research in chemical graph theory. These indices are numerical values associated with compounds that aim to establish connections between chemical structures and physical attributes, chemical reactivity, and biological activity. Many graph polynomials have been proposed to compute various topological indices. Quantitative structure-property relationship (QSPR) analysis of chemical compounds involves several regression methods that rely on these topological indices. In this article, we utilize the newly introduced DƐ-polynomial to calculate the eccentricitybased indices such as the diameter eccentricity based and hyper diameter eccentricity based index values for nineteen linear saturated monocarboxylic acids. Quantitative structureproperty relationship (QSPR) analysis is also performed for the seven thermodynamic properties of monocarboxylic acids. The relationship between thermodynamic properties and topological indices is explored using linear, quadratic and cubic regression models. The best fit curvilinear regression models were determined based on the R2 and RMSE values of the models. To further investigate the statistical significance of our models, we also conducted chi-square goodness-of-fit tests to identify the best fit models.

Role of GA, AG and R in Structure-Property Modelling

Molecular descriptors are essential tools that bridge the gap between chemical structures and their properties. Among these, the Randić index (R), geometric-arithmetic index (GA), and arithmetic-geometric index (AG) are well-established. However, the relationships between these descriptors, particularly the gaps GA − R and AG − R , have not been extensively explored. This study investigates the impact of these descriptor gaps on chemical graph theory. Our findings reveal that the differences GA − R and AG − R provide unique insights into the structure-property modeling of molecules, particularly outperforming AG, GA, R, and other known descriptors for alkanes. Polycyclic aromatic compounds (PACs) are organic molecules composed of multiple fused aromatic rings. PACs are of significant environmental concern due to their persistence and potential toxicity, including carcinogenic properties. The effectiveness of GA − R and AG − R is observed to be strong in structure-property modeling for some polycyclic aromatic compounds. Additionally, GA − R and AG − R are recognized as important descriptors for various pharmaceutical compounds employed in treating malignant diseases.

Two-Matchings with Respect to the General Sum-Connectivity Index of Trees

A vertex-degree-based topological index φ associates a real number to a graph G which is invariant under graph isomorphism. It is defined in terms of the degrees of the vertices of G and plays an important role in chemical graph theory, especially in QSPR/QSAR investigations. A subset of k edges in G with no common vertices is called a k-matching of G, and the number of such subsets is denoted by mG,k. Recently, this number was naturally extended to weighted graphs, where the weight function is induced by the topological index φ. This number was denoted by mkG,φ and called the k-matchings of G with respect to the topological index φ. It turns out that m1G,φ=φG, and so for k≥2, the k-matching numbers mkG,φ can be viewed as kth order topological indices which involve both the topological index φ and the k-matching numbers. In this work, we solve the extremal value problem for the number of 2-matchings with respect to general sum-connectivity indices SCα, over the set Tn of trees with n vertices, when α is a real number in the interval −1,0.

On statistical evaluation of reverse degree based topological indices for iron telluride networks

In the context of graph theory and chemical graph theory, this research conducts a detailed mathematical investigation of reverse topological indices as they relate to iron telluride networks, clarifying their complex interactions. Graph theory is a branch of abstract mathematics that carefully studies the connections and structural features of graphs made up of edges and vertices. These theoretical ideas are expanded upon in chemical graph theory, which models molecular architectures with atoms acting as vertices and chemical bonds as edges. By extending these concepts, this work investigates the reverse topological indices in the context of Iron Telluride networks and outlines their significant effects on chemical reactivity, molecular topology and statistical modeling. By navigating intricate mathematical formalisms and algorithmic approaches, the analysis provides profound insights into the reactivity patterns and structural dynamics of Iron Telluride compounds, enhancing our knowledge of solid-state chemistry and materials science.

Enumeration of the Hosoya index of pericondensed benzenoid system

In chemical graph theory, topological indices play an important role and have various uses in quantitative structure-property relationships (QSPR) as well as quantitative structure-activity relationships (QSARs). The Hosoya index is one of them, performing a molecular descriptor in mathematical chemistry. Therefore, progressing with investigations requires computing the Hosoya index of distinct molecular graphs. This research presents a computational method for determining the Hosoya index of pericondensed benzenoid systems using the transfer matrix approach and the Hosoya vector.

Modern chemical graph theory

AbstractGraph theory has a long history in chemistry. Yet as the breadth and variety of chemical data is rapidly changing, so too do graph encoding methods and analyses that yield qualitative and quantitative insights. Using illustrative cases within a basic mathematical framework, we showcase modern chemical graph theory's utility in Chemists' analysis and model development toolkit. The encoding of both experimental and simulation data is discussed at various levels of granularity of information. This is followed by a discussion of the two major classes of graph theoretical analyses: identifying connectivity patterns and partitioning methods. Measures, metrics, descriptors, and topological indices are then introduced with an emphasis upon enhancing interpretability and incorporation into physical models. Challenging data cases are described that include strategies for studying time dependence. Throughout, we incorporate recent advancements in computer science and applied mathematics that are propelling chemical graph theory into new domains of chemical study.This article is categorized under: Molecular and Statistical Mechanics > Molecular Dynamics and Monte‐Carlo Methods Structure and Mechanism > Computational Materials Science Structure and Mechanism > Molecular Structures

Computational insights into zinc silicate MOF structures: topological modeling, structural characterization and chemical predictions

Metal-organic frameworks (MOFs) play a pivotal role in modern material science, offering unique properties such as flexibility, substantial pore space, distinctive structure, and large surface area. Recently, zinc-based MOFs have attracted significant attention, particularly in the biomedical arena, owing to their versatile applications in drug delivery, biosensing, and cancer imaging. However, there remains a crucial need to explore and understand the structural properties of zinc silicate-based MOFs to fully exploit their potential in various applications. The objective of this study is to address this need by employing topological modeling techniques to characterize zinc silicate networks. Utilizing connection number concept of chemical graph theory and novel AL molecular descriptors, we aim to investigate the structural intricacies of these MOFs. More precisely, zinc silicate-based MOF networks are topologically modeled via novel AL topological indices, and derived mathematical closed form formulae for them. By comparing experimental and calculated values and constructing linear regression models, the predictive capabilities of the proposed descriptors are evaluated. Specifically, the performance of derived topological indices against the physico-chemical properties of octane isomers is assessed, which provide valuable insights into their predictive potential. The findings of this study demonstrated the potential of novel AL indices in predicting a wide range of important physico-chemical properties, further enhancing their practicality in materials science and beyond.

On physical analysis of phthalocyanine iron (II) using topological descriptor and curve fitting models

A new area of applied chemistry called chemical graph theory uses combinatorial techniques to explain the complex interactions between atoms and bonds in chemical systems. This work investigates the use of edge partitions to decipher molecular connection patterns. The main goal is to use topological indices that capture important topological features to create a connection between the thermodynamic properties and structural characteristics of chemical molecules. We specifically examine the complex web of atoms and links that make up the Fe phthalocyanine chemical graph. Moreover, our study demonstrates a relationship between the calculated topological indices and the thermodynamic properties of Fe phthalocyanine (Phthalocyanine Iron (II)). This work offers insight into the thermodynamic consequences of molecule structures. It advances the subject of chemical graph theory, providing a useful perspective for future applications in catalysis and materials science.

Topological Characterization for Triangular, RegularTriangular Oxides and Silicates Networks

Chemical graph theory can be studied with the aid of mathematical tools called m-polynomials. M-Polynomials offer a potent tool for computing different topological indices associated with vertex degrees and analyzing degree-based structural information in graphs. By counting specific substructure types within them, they are able to encode information about the structure of molecules or networks. In this article, we have developed M-Polynomials with the help of different topological invariants such as first Zagreb (M1(β)), second Zagreb (M2(β)), second modified Zagreb (Mm2(β)), inverse sum (I(β)), harmonic index (H(β)) and Randic index (Rα0(β)) for the molecular structures of Triangular oxide TOX(r), Regular triangular oxide RTOX(r), Triangularsilicate TSL(r) & Regular triangular silicate RTSL(r) networks to introduce new closed formulas to get better understanding the applications of M-Polynomials and topological indices in mathematical chemistry especially in the field of QSAR and QSPR study with the help of some software like MATLAB. We have also discussed the graphical behaviors of the above-mentioned structures.

Edge dependent fault-tolerance in certain carbon-based crystal structures

The field of chemical graph theory encompasses interdisciplinary studies that utilize mathematical methods in exploring the attributes and arrangement of chemical compounds. Chemical graphs, in which atoms are vertices and bonds are edges, offer influential mathematical representations for portraying the molecular structures of chemical compounds. A smallest subset of vertices of a graph from which the vector of distances to each vertex of the graph is unique is known as metric basis for that graph and cardinality of a metric basis is known as metric dimension of that graph. In this paper, an important variant of metric dimension known as fault-tolerant edge metric dimension (FTEMD) is taken into consideration and computed it for most important possible allotrope of carbon family, known as crystal cubic carbon (denoted by G(n)). We prove that the FTEMD value of the molecular graph of crystal cubic carbon is unbounded for large n. Also, the comparison between several variants of metric dimension and FTEMD for the molecular graph G(n) has also been incorporated in the paper.

On spectral radius of generalized arithmetic–geometric matrix of connected graphs and its applications

The generalized arithmetic–geometric matrix of a simple connected graph G=(V,E) is defined to be the |V|×|V| matrix whose ij-entry is diα+djα2diαdjα if vivj∈E, and 0 otherwise, where α is an arbitrary real number and di is the degree of vi∈V. This matrix is a general form of the arithmetic–geometric matrix (α=1) and the extended adjacency matrix (α=2), both of which have been well studied in spectral graph theory and chemical graph theory. In this paper, we focus on the spectral radius ρagα of the generalized arithmetic–geometric matrix of connected graphs. Some chemical applications of ρagα are explored, and some extremal results on ρagα are obtained; in particular, the connected graphs (including trees) having the maximum and minimum ρagα are determined.

Reverse Zagreb Indices and Their Application in the Evaluation of Physiochemical Properties of Anticancer/Antibacterial Drugs.

Recent advancements in chemical graph theory have facilitated a deeper understanding of the relationship between chemical structures and their underlying graphs based upon classical graph theory principles. Quantitative structure-property relationship (QSPR) analysis stands out as a powerful tool for probing chemical graphs. Topological indices, graph invariants assigning numerical values to graphs, play a pivotal role in statistically correlating physical properties, chemical reactivity, and biological activity across diverse chemical structures. The cactus graph (CG) represents a noteworthy family of connected graphs distinguished by the absence of common vertices between cycles. In this study, we focus on establishing the expressions of Zagreb and reverse Zagreb indices in terms of known parameters for cactus graphs. Subsequently, we leverage these indices to conduct QSPR analysis, employing linear and multilinear regression models to investigate the relationship between the properties of chemical structures adorned with cactus-like graphs and the corresponding Zagreb and reverse Zagreb indices.

On Certain Degree Based and Bond-additive Topological Indices of Dodeca-benzo-circumcorenene.

Chemical graph theory has been used to mathematically model the various physical and biological aspects of chemical substances. A mathematical formulation that may be applied to any graph and can characterise a molecule structure is known as a topological index or molecular descriptor. It is convenient and efficient to analyse the mathematical values and further research on various physical properties of a molecule based on these molecular descriptors. They provide useful alternatives to lengthy, expensive, and labour-intensive laboratory experiments. The topological indices can be used to predict the chemical structures, physicochemical properties, and biological activities using quantitative structure-activity relationships (QSARs) and quantitative structure-property relationships (QSPRs). In this study, the molecular descriptors of the Dodeca-benzo-circumcorenene compounds are derived based on their corresponding molecular structures. The computed indices are then compared graphically to study their relationship with the molecular structure and with each other..

Topological properties of fractals via M-polynomial

Sierpiński graphs are frequently related to fractals, and fractals apply in several fields of science, i.e., in chemical graph theory, computer networking, biology, and physical sciences. Functions and polynomials are powerful tools in computer mathematics for predicting the features of networks. Topological descriptors, frequently graph constraints, are absolute values that characterize the topology of a computer network. In this essay, Firstly, we compute the M-polynomials for Sierpiński-type fractals. We derive some degree-dependent topological invariants after applying algebraic operations on these M-polynomials.

New Formulas and New Bounds for the First and Second Zagreb Indices of Phenylenes

Graph theory is widely used to represent and analyze chemical structures. In addition, topological indices developed for graphs have a connection with the relationships of chemical structures such as physicochemical and bioactivity. Topological indices are widely used in QSPR-QSAR analysis and have found many applications in chemical graph theory. The oldest known degree-dependent topological indices are the first and second Zagreb indices. These indices have found wide application in chemical structures. Phenylenes containing aromatic and antiaromatic rings exhibit unique physicochemical properties and there is a wide variety of studies on phenylenes. In this article, we present some new formulas and lower bounds for the first and second Zagreb indices molecular structures of phenylenes. In addition, the BFS algorithm method, which is one of the graph algorithms, was used for the first time for the boundary study of chemical structures.

Exploring expected values of topological indices of random cyclodecane chains for chemical insights

Chemical graph theory has made a significant contribution to understand the chemical compound properties in the modern era of chemical science. At present, calculation of the topological indices is one of most important area of research in the field of chemical graph theory. Cyclodecane is a cyclic hydrocarbon with the chemical formula C10H20\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C_{10}H_{20}$$\\end{document}. It consists of a ring of ten carbon atoms bonded together in a cyclical structure. Cyclodecane chains can be part of larger molecules or polymers, where multiple cyclodecane rings are connected together. These molecules can have various applications in chemistry, materials science, and pharmaceuticals. This article aims to determine expected values of some connectivity based topological indices of random cyclodecane chains, containing saturated hydrocarbons with at least two rings. It also compares these descriptors using explicit formulae, numerical tables and present graphical profiles of these comparisons.

Forgotten Topological and Wiener Indices of Prime Ideal Sum Graph of Zn.

Chemical graph theory is a sub-branch of mathematical chemistry, assuming each atom of a molecule is a vertex and each bond between atoms as an edge. Owing to this theory, it is possible to avoid the difficulties of chemical analysis because many of the chemical properties of molecules can be determined and analyzed via topological indices. Due to these parameters, it is possible to determine the physicochemical properties, biological activities, environmental behaviours and spectral properties of molecules. Nowadays, studies on the zero divisor graph of Z_n via topological indices is a trending field in spectral graph theory. For a commutative ring R with identity, the prime ideal sum graph of R is a graph whose vertices are nonzero proper ideals of R and two distinctvertices I and J are adjacent if and only if I+J is a prime ideal of R. In this study the forgotten topological index and Wiener index of the prime ideal sum graph of Z_n are calculated for n=p^α,pq,p^2 q,p^2 q^2,pqr,p^3 q,p^2 qr,pqrs where p,q,r and s are distinct primes and a Sage math code is developed for designing graph and computing the indices. In the light of this study, it is possible to handle the other topological descriptors for computing and developing new algorithms for next studies and to study some spectrum and graph energies of certain finite rings with respect to PIS-graph easily.