Atom-bond Connectivity Research Articles

Topological Characterization of Coronoid Structures via Valency Based Topological Indices.

The number of atoms producing a bond with an atom a in a molecular structure (graph) is regarded as the valency of a. A number of molecular structures have been topologically characterized via a large number of valency-based topological indices, which correlate certain physio-chemical properties like boiling point, stability, strain energy, and many more chemical compounds. Our intention is to perform the topological characterization of primitive coronoid structures initially on the base of seven different valency-based topological indices. Fundamentally, our purpose is to provide the universal formula by using the topological characterization of any primitive coronoid structures with any valency-based topological index. The technique of atom-bonds partitions according to the valency of atoms besides counting principals are used to find the results. Exact values of valency-based topological index, namely the Randic index, sum connectivity index, harmonic index, atom bond connectivity index, geometric arithmetic index, forgotten index, and symmetric division index are calculated coronoid structures initially constructed with at most three benzene lyres. All these indices are valency-dependent structural invariants that have been used to elaborate several chemical properties of compounds of the structure. Moreover, we provide the general formula to compute any valency-based topological index for any type of coronoid structure constructed with any number of benzene lyres.

Predictive potential of distance-related spectral graphical descriptors for structure-property modeling of thermodynamic properties of polycyclic hydrocarbons with applications

A distance-related spectral descriptor is a graphical index with defining structure built on eigenvalues of chemical matrices relying on distances in graphs. This paper explores the predictive ability of both existing and new distance-related spectral descriptors for estimating thermodynamic characteristics of polycyclic hydrocarbons (PHs). As a standard choice, the entropy and heat capacity are selected to represent thermodynamic properties. Furthermore, 30 initial members of PHs are considered as test molecules for this study. Three new molecular matrices have been proposed and our research demonstrates that distance-spectral graphical indices built by these novel matrices surpass in efficiency relative to famous distance-spectral indices. First, a novel computational method is put forwarded to evaluate distance-spectral indices of molecular graphs. The proposed methodology is utilized to compute both pre-existing and novel distance-related spectral descriptors, with an aim to assess their predictive efficacy using experimental data pertaining to two selected thermodynamic properties. Subsequently, we identify the five most promising distance-related spectral descriptors, comprising the degree-distance and Harary energies, the recently introduced second geometric-arithmetic energy along with its associated Estrada invariant, and 2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ ext {nd}$$\\end{document} atom-bond connectivity (ABC) Estrada index. Notably, the 2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ ext {nd}$$\\end{document} ABC Estrada index and Harary energy demonstrate correlation coefficients exceeding 0.95, while certain conventional spectral indices including the distance energy as well as its associated Estrada index, display comparatively lower performance levels. Moreover, we illustrate the practical implications of our findings on specific classes of one-hexagonal nanocones and carbon polyhex nanotubes. These outcomes hold potential for enhancing the theoretical determination of certain thermodynamic attributes of these nanostructures, offering improved accuracy and minimal margin of error.

On tricyclic graphs with maximum atom–bond sum–connectivity index

The sum-connectivity, Randić, and atom-bond connectivity indices have a prominent place among those topological indices that depend on the graph's vertex degrees. The ABS (atom-bond sum-connectivity) index is a variant of all the aforementioned three indices, which was recently put forward. Let T(n) be the class of all connected tricyclic graphs of order n. Recently, the problem of determining graphs from T(n) having the least possible value of the ABS index was solved in (Zuo et al., 2024 [39]) for the case when the maximum degree of the considered graphs does not exceed 4. The present paper addresses the problem of finding graphs from T(n) having the largest possible value of the ABS index for n≥5.

Topological characterization of PETAA and bismuth(III) iodide: Flair drug carriers and therapeutic agents

Dendrimers recently emerged as a strong category of nanoparticles in the realm of nanomedicine due to their distinct structural properties. In order to bring the medications to mark the carrier vehicle, a specific design with the right qualities (topological invariants) might be targeted using the physical chemistry of dendrimers. Topological descriptors of chemical networks are numeric numbers that allow us to obtain information about a compound’s structure and expose its underlying properties without the requirement for validation. In this work, we have computed the atom-bond connectivity index (ABC) and the atom-bond sum-connectivity index (ABS) of bismuth(III) iodide and PETAA dendrimer. A graphical comparison is given at the end to study the behavior of these two topological indices for bismuth (III) iodide and PETAA dendrimer.

On Ve-Degree and Ev-Degree Based Topological Invariants of Chemical Structures

In the realm of graph theory, recent developments have introduced novel concepts, notably the ν ε -degree and ε ν -degree, offering expedited computations compared to traditional degree-based topological indices (TIs). These TIs serve as indispensable molecular descriptors for assessing chemical compound characteristics. This manuscript aims to meticulously compute a spectrum of TIs for silicon carbide S i C 4 - I [ r , s ] , with a specific focus on the ε ν -degree Zagreb index, the ν ε -degree Geometric-Arithmetic index, the ε ν -degree Randić index, the ν ε -degree Atom-bond connectivity index, the ν ε -degree Harmonic index, and the ν ε -degree Sum connectivity index. This study contributes to the ongoing advancement of graph theory applications in chemical compound analysis, elucidating the nuanced structural properties inherent in silicon carbide molecules.

Further Results for Some Eccentric Based Indices.

Topological indices have a significant role in determining a molecule's topology in the light of theoretical chemistry. This study aimed to find new bounds for some eccentric-based topological indices. We used comparison theorem for integrals. We found new bounds for the eccentric-based harmonic index and atom-bond connectivity index. Our results are better than previous results in the literature. This study has contributed to the chemical graph theory literature by giving a new method to find eccentric-based topological indices with lower and upper bounds.

Atom-bond-connectivity (ABC) indices of graphene sheets, zigzag single walled carbon nanotubes and single walled carbon nanotori

Abstract Atom-bond-connectivity (ABC) indices are obtained in analytical forms for graphene sheets, zigzag single walled carbon nanotubes (SWCNTs), and single walled carbon nanotori in terms of number of rings (r) that measures the length and the number of hexagons in between two rings (h) that dictates the width of the concerned systems. The procedures followed for ABC index have been used to obtain the expressions of augmented Zagreb and Randić indices for such systems. Logarithm of ABC indices of zigzag SWCNTs are found to correlate linearly well with the bond dissociation energies per C–C bond and the Young’s moduli of said SWCNTs with fixed number of rings (r) but varying number of hexagons (h) in between two successive rings. The plot of logarithm of ABC index versus Young’s modulus of such SWCNTs in varying both r and h simultaneously is not a straight line but fits well with the sigmoidal (Boltzmann) curve. Wiener index, one of the important distance based index, has recently been found to have similar correlations with the concerned properties of such systems. Similar plots would appear for the said properties of the zigzag SWCNTs with other degree-based indices like augmented Zagreb and Randić indices, as have been indicated from their respective expressions obtained.

Reduced reverse degree-based topological indices of graphyne and graphdiyne nanoribbons with applications in chemical analysis

Graphyne and Graphdiyne Nanoribbons reveal significant prospective with diverse applications. In electronics, they propose unique electronic properties for high-performance nanoscale devices, while in catalysis, their excellent surface area and reactivity sort them valuable catalyst supports for numerous chemical reactions, contributing to progresses in sustainable energy and environmental remediation. The topological indices (TIs) are numerical invariants that provide important information about the molecular topology of a given molecular graph. These indices are essential in QSAR/QSPR analysis and play a significant role in predicting various physico-chemical characteristics. In this article, we present a formula for computing reduced reverse (RR) degree-based topological indices for graphyne and graphdiyne nanoribbons, including the RR Zagreb indices, RR hyper-Zagreb indices, RR forgotten index, RR atom bond connectivity index, and RR Geometric-arithmetic index. We also execute a graph-theoretical analysis and comparison to demonstrate the critical significance and validate the acquired results. Our findings provide insights into the structural and chemical properties of these nanoribbons and contribute to the development of new materials for various applications.

On some extended energy of graphs and their applications

In this paper, we established a relation between the extended energy of graphs such as the first and second Zagreb energy, Randi?c energy, reciprocal Randi?c energy, and the atom-bond connectivity energy with some thermodynamic properties of benzenoid hydrocarbons. We have seen that these indices are well correlated to the boiling point (BP), Kovats retention index (RI), entropy (S), enthalpy of formation (?Hf), octanol-water partition coefficient (logP), and acentric factor (?) of benzenoid hydrocarbons.

Entropy measures of the metal–organic network via topological descriptors

Abstract A family of chemical compounds known as metal–organic networks (MONs) is composed mainly of clusters of metal ions with organic ligands. It can increase volatility or make substances soluble in organic solvents. By using these salient features, organic compounds generate applications in material sciences for sol–gel processing. A graph’s entropy is utilized as a complexity indicator and is interpreted as the structural information content of the graph. Investigating the entropies of relationship systems is a common occurrence in discrete mathematics, computer science, information theory, statistics, chemistry, and biology. In this article, we investigated the degree-based entropies: geometric arithmetic entropy, atom bond connectivity entropy, general Randic′ entropy, and general sum connectivity entropy for MONs. Furthermore, we created tables for all expressions by using 1–10 values for the s s parameter of these entropies.

On some topological indices of zero divisor graphs of direct product of three finite fields

In this paper, we determine some degree-based topological indices, such as the Sombor index, the first and second Zagreb indices, the forgotten topological index, the Narumi–Katayama index, the first and second multiplicative Zagreb indices, the atom-bond connectivity index, and eccentricity-based topological indices such as total eccentricity, the first and second Zagreb eccentricity indices, and the eccentricity connectivity index of the zero divisor graph with vertex set non-zero zero divisors of the reduced ring of the direct product of three finite fields.

Extremal problems on the Atom-bond sum-connectivity indices of trees with given matching number or domination number

The atom-bond sum-connectivity (ABS) index of a graph is a variant from some famous chemical topological indices such as the Randić index, the sum-connectivity index and the atom-bond connectivity index. The research on its extremal problems of a graph has much theoretical value and application background.Let Tn,m and T(n,γ) be the sets of all trees on n vertices with given matching number m and given dominating number γ, respectively. In this paper, we firstly determine the sharp upper bound of the ABS index among Tn,m and characterize the corresponding extremal graph. Secondly, we determine the sharp upper and lower bounds of the ABS index among T(n,γ) by using the bridge of the matching theory. Finally, the corresponding topological structures of their extremal graphs are characterized, respectively.

On the Difference of Atom-Bond Sum-Connectivity and Atom-Bond-Connectivity Indices

The atom-bond connectivity (ABC) index is one of the wellinvestigated degree-based topological indices. The atom-bond sumconnectivity (ABS) index is a modified version of the ABC index, which was introduced recently. The primary goal of the present paper is to investigate the difference between the aforementioned two indices, namely ABS − ABC. It is shown that the difference ABS − ABC is positive for all graphs of minimum degree at least 2 as well as for all line graphs of those graphs of order at least 5 that are different from the path and cycle graphs. By means of computer search, the difference ABS − ABC is also calculated for all trees of order at most 15.

Connection number topological aspect for backbone DNA networks.

The present study investigates the complex topological characteristics of DNA networks, with a specific emphasis on the innovative metric known as Connection Number (CN) as a key factor in determining network structure. The Connection Number, represented as CN(v) for a vertex v, measures the count of unique paths that link v to every other vertex in the network. By employing rigorous mathematical modeling and analysis techniques, we are able to reveal the profound implications of CN (complex networks) in characterizing the structural robustness and dynamics of information flow within DNA networks. The study of how the theory of graphs and chemicals interact is known as chemical graph theory. This paper, computing the hyper Zagreb connection index, augmented connection index, inverse sum connection index, harmonic connection index, symmetric division connection index, geometric arithmetic connection index, and atom bond connectivity connection index, of two significant types of backbone DNA and Barycentric subdivision of backbone DNA networks. Direct method computation is used to produce these Connection-based topological descriptors.

Degree and distance based topological descriptors of power graphs of finite non-abelian groups

A topological descriptor is a numerical value derived from the molecular structure that encapsulates the most important structural characteristics of the molecule under consideration. Fundamentally, it involves assigning an algebraic value to the composition of chemicals while developing a relationship between this value and several physical properties, like biological activity, and chemical reactivity. This article examines multiple kinds of degree and eccentricity-based topological indices for power graphs of various finite groups. We calculate the Wiener index and its reciprocal, atom-bond connectivity index and its fourth version, the Schultz index, the geometric–arithmetic and harmonic indices, and finally determine the general Randić and Harary indices of power graphs of finite cyclic and non-cyclic groups of order pq, dihedral, and generalized quaternion groups, where p,q(q≥p) are distinct primes.

Mapping Connectivity Patterns: Degree-Based Topological Indices of Corona Product Graphs

Graph theory (GT) is a mathematical field that involves the study of graphs or diagrams that contain points and lines to represent the representation of mathematical truth in a diagrammatic format. From simple graphs, complex network architectures can be built using graph operations. Topological indices (TI) are graph invariants that correlate the physicochemical and interesting properties of different graphs. TI deal with many properties of molecular structure as well. It is important to compute the TI of complex structures. The corona product (CP) of two graphs G and H gives us a new graph obtained by taking one copy of G and V G copies of H and joining the i th vertex of G to every vertex in the i th copy of H . In this paper, based on various CP graphs composed of paths, cycles, and complete graphs, the geometric index (GA) and atom bond connectivity (ABC) index are investigated. Particularly, we discussed the corona products P s ⨀ P t , C t ⨀ C s , K t ⊙ K s , K t ⊙ P s , and P s ⊙ K t and GA and ABC index. Moreover, a few molecular graphs and physicochemical features may be predicted by considering relevant mathematical findings supported by proofs.

On topological indices of some chemical graphs

This research delves into the application of topological indices in predicting the physico-chemical properties of chemical materials. These indices have been employed in various QSAR and QSPR studies, providing a mathematical formulation to predict properties such as viscosity and radius of gyrations Rucker and Rucker [J. Chem. Inf. Comput. Sci. 39, 788–802, (1999)]. We explore several indices, including the forgotten topological index, inverse sum index, general sum connectivity index and first multiplicative atom bond connectivity index, and investigate their usefulness in predicting bio-activity and anti-inflammatory activities of certain chemical compounds. Specifically, we focus on the allotrope structure, nanotubes and dominating oxide and silicate chemical networks. We present a graphical trend of these indices and perform a comparison to facilitate the observation of physical-chemical properties of these materials for researchers.

Topological Descriptors of Molecular Networks via Reverse Degree

Porphyrazine and tetrakis porphyrazine are organic compounds with intricate ring structures. They are made up of four indole rings that fuse together to form a larger cyclic structure. These structures have been extensively studied in physics and chemistry. They exhibit unique electronic and optical properties due to their highly conjugated systems. In several scientific domains, they are used as the building blocks for functional materials, molecular electronics, sensors, and catalysis. In order to derive a topological index from data, molecules are represented as graphs, with atoms acting as nodes and bonds acting as edges. There are many different techniques and algorithms that can be used to determine the topological index. We have used the reverse degree-based methodology for the mentioned structures. Reverse degree-based topological indices evaluate the structural complexity and stability of chemical compounds by adding the reversed atom degree sequence to their molecular networks. In this study, we calculated the reverse degree-based topological indices like the reverse redefined Zagreb indices, the reverse forgotten index, the reverse hyper-Zagreb index, the reverse Balaban index, the reverse atom bond connectivity index, the reverse geometric arithmetic index, and the reverse general Randić index for porphyrazine and tetrakis porphyrazine networks. This research advances our knowledge of the fundamental laws of physics and chemistry by illuminating the complex interrelationships between molecular structure, biological interactions, and chemical reactivity.

Analysis of the properties of the topological Index using (analysis tools)

Graph G has two sets of information: the vertices, V(G), and the edges, E(G). The definitions for the Connectivity, Geometric Arithmetic, Atomic Bond, and Sum Connectivity Indices of G:
 
 were deg(u), deg(v) are a degree of vertices. Dendrimers are synthetic, man-made molecules that are composed of monomers organized in a branching structure. in this article, we calculate Connectivity, Geometric Arithmetic, Atom Bond Connectivity, and Sum Connectivity Index for the PAMAM, POPAM, and HACN1J dendrimers.

On bond incident degree index of chemical trees with a fixed order and a fixed number of leaves

A tree in which no vertex has a degree greater than 4 is called a chemical tree. The bond incident degree index of a chemical tree T is defined as ∑xy∈ETφ(degT(x),degT(y)), where ET is the edge set of T, φ is a real-valued symmetric function, and degT(x) stands for the degree of a vertex x of T. This paper reports extremal results on bond incident degree indices of chemical trees with a fixed order and a fixed number of leaves. Furthermore, we use these results directly to some renowned topological indices, such as symmetric division deg index, Randić index, geometric-arithmetic index, sum-connectivity index, Sombor index, harmonic index, multiplicative sum Zagreb index, atom-bond connectivity index, etc.