Chemical Graph Theory Research Articles

Analyzing boron oxide networks through Shannon entropy and Pearson correlation coefficient

In the current age of chemical science, chemical graph theory has significantly advanced our understanding of the characteristics of chemical compounds. To simulate the mathematical, chemical, and physical aspects of networks, a topological index, a numerical measure obtained from the graph of a chemical network, employed. Recent work has explored the topological properties of boron oxide using chemical graph theory. In this work, we conduct a Pearson correlation analysis of boron oxide to assess the correlations between the Van and S indices and entropy metrics. We analyze the Pearson correlation coefficients between the entropy values and the calculated indices using a heatmap. In this article, a significant positive correlation between the Van, and S indices, and entropy values, which is represented by the heatmap of the strong linear correlations. To avoid duplication, a dimensionality reduction technique should be used for highly connected variables. Additionally, this study gives a detailed explanation of the link between the indices and entropy, which will form the basis of further statistical investigations.

Application of the Nirmala Index in Nanotechnology: Optimizing Molecular Structures for Advanced Nanomaterials

Abstract Chemical graph theory has played a key role in advancing our understanding of molecular structure by producing degree-based topological indices that forecast crucial physical and chemical properties. In this paper we attempt to investigate this recently defined Nirmala index as an invariant from different topological angles and applications on multiple molecular graph structures such as triangular, double, and alternate quadrilateral snake structures. This index offers a more profound comprehension of the connections between molecular structures and attributes. The next generation of ground breaking developments in nanotechnology, where the creation of devices and functional nanomaterials hinges on molecular interaction at the nanoscale, may also be greatly influenced by this index. These indices have potential applications in drug delivery systems and in molecular structure optimization related to nanotechnological fields like nanophotonics and nanoelectronics. Consequently, the Nirmala index will pave the way for the development of more sophisticated tools for the production of high-performing nanoscale materials.

Analyzing Single-Valued Neutrosophic Fuzzy Graphs Through Matroid Perspectives

We hope to present this paper on the emergence of a novel category of matroids derived from single-valued neutrosophic (SVN) fuzzy-graphs. The findings of this study make a substantial contribution to both matroid theory and the field of neutrosophic systems. In this study, we aim to introduce the concept of SVN-matroids and conduct a thorough investigation into their fundamental properties. Additionally, we explore the essential requirements for establishing duality within the realm of matroids formed from neutrosophic fuzzy-networks. This investigation encompasses co-weak isomorphism, weak isomorphism, and isomorphism between two matroids. Furthermore, our research proposes the expansion of other graph products and the creation of a SVN-graph to utilize the Greedy and Kruskal algorithms for determining optimal productivity in an information flow of a single-source network within a SVN-network. The application of neutronosophic graph theory offers a paradigm for addressing optimization problems in chemical graph theory, such as identifying the maximal independent set and a minimum spanning tree. When faced with uncertainty, neutrosophic optimization provides more reliable solutions by considering the uncertainties associated with the objective function or constraints.

Weighted Asymmetry Index: A New Graph-Theoretic Measure for Network Analysis and Optimization

Graph theory is a crucial branch of mathematics in fields like network analysis, molecular chemistry, and computer science, where it models complex relationships and structures. Many indices are used to capture the specific nuances in these structures. In this paper, we propose a new index, the weighted asymmetry index, a graph-theoretic metric quantifying the asymmetry in a network using the distances of the vertices connected by an edge. This index measures how uneven the distances from each vertex to the rest of the graph are when considering the contribution of each edge. We show how the index can capture the intrinsic asymmetries in diverse networks and is an important tool for applications in network analysis, optimization problems, social networks, chemical graph theory, and modeling complex systems. We first identify its extreme values and describe the corresponding extremal trees. We also give explicit formulas for the weighted asymmetry index for path, star, complete bipartite, complete tripartite, generalized star, and wheel graphs. At the end, we propose some open problems.

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.

Some Reverse Topological Indices of Comet and Double Comet Graphs

Introduction: Chemical Graph Theory or CGT for short, is a transdisciplinary field of Mathematics wherein graphs are used to represent chemical compounds. Under this representation, the atoms of a chemical compound are expressed as vertices, while the bonds connecting these atoms are expressed as edges. Once the graph representation of a chemical compound has been determined, graph-theoretic techniques can now be used to determine various topological indices associated with the chemical compound. Topological indices are invariants of a graph that have predictive power for the chemical properties of a chemical compound. In CGT, trees, being connected and acyclic, have been an essential class of graphs. This is because, most compounds have acyclic molecular structures. In the 2023 study by Gowtham and Husin, various reverse topological indices of a family of trees called bistar graphs have been determined. Objectives: Motivated by the work of Gowtham and Husin, we determine some reverse topological indices of another family of trees called comets and double comets. Double comets are natural extension of bistar graphs. We also give a certain computational application of our results on double comet graphs. Finally, we investigate the relationship between the reverse topological indices of bistar graphs and double comets. Methods: Several important graph-theoretic concepts were used to analyze some important properties of comets and double comets, leading to the computation of their reverse vertex degree based topological indices. Results: The following reverse vertex degree-based topological indices for comets and double comets were determined: Reverse sum-connectivity, First reverse Zagreb, Second reverse Zagreb, Reverse arithmetic-geometric, Reverse geometric-arithmetic, Reverse Sombor, and Reverse Nirmala. A computational application of the results to a certain chemical compound (2,2,4,4-Tetramethylpentane) or C9H20 were also demonstrated. Conclusions: The results of the paper provided an extension to an existing result on the reverse vertex degree-based topological indices of bistar graphs by considering double comet graphs. Several topological reverse vertex degree-based topological indices were also computed for comet graphs. The computed topological indices were also applied in determining some invariants of a certain chemical compound. For future studies, we recommend that reverse vertex degree based topological indices of other family of trees will be considered.

Topology of quasi divisor graphs associated with non-associative algebra

The visualization of graphs representing algebraic structures has increasingly gained traction in chemical engineering research, emerging as a significant scientific challenge in contemporary studies. Due to their practical applications, graphs associated with groups and various algebraic systems have garnered significant interest in academic literature. Chemical graph theory employs graphs to illustrate molecular structures, where vertices symbolize atoms and edges represent bonds. The structure of quasi divisor graphs linked to non-associative algebra, including quasigroups and loops, presents numerous potential implications and wider applications. These applications span multiple disciplines, such as algebra, combinatorics, computer science, and network theory. Latin squares are tabular representations of quasigroups, generalizations of group structure. Numerous classifications of quasigroups exist in academic literature, such as Moufang quasigroups, Bol quasigroups, and alternative quasigroups. However, the structural characteristics of quasigroups exhibiting the weak inverse property closely resemble those of groups. The overall purpose of this study is to see the interrelationship between non-isomorphic quasi divisor graphs and a particular class of G-loops. To achieve our results, we employ a variety of methods, including a combinatorial approach, degree counting techniques, vertex and edge partitioning methods, as well as graph-theoretical tools and analytical procedures. In this paper, we compute a few topological indices based on degree, distance, and degree-distance using the structural aspects of quasi divisor graphs of orders 2φ,22ϕ,23ϕ and 4ϕ1ϕ2 where φ is a positive integer and ϕ1,ϕ2 are odd primes with ϕ1<ϕ2. This work also includes polynomial forms and their geometrical interpretations connected to linear, quadratic and higher degree topological indices of quasi divisor graphs associated with a class of non-commutative weak inverse property quasigroups.

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 &gt; Molecular Dynamics and Monte‐Carlo Methods Structure and Mechanism &gt; Computational Materials Science Structure and Mechanism &gt; 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) &amp; 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.