Geometric-arithmetic Index Research Articles

On Unicyclic Graphs with a Given Number of Pendent Vertices or Matching Number and Their Graphical Edge-Weight-Function Indices

Consider a unicyclic graph G with edge set E(G). Let f be a real-valued symmetric function defined on the Cartesian square of the set of all distinct elements of G’s degree sequence. A graphical edge-weight-function index of G is defined as If(G)=∑xy∈E(G)f(dG(x),dG(y)), where dG(x) denotes the degree a vertex x in G. This paper determines optimal bounds for If(G) in terms of the order of G and a parameter z, where z is either the number of pendent vertices of G or the matching number of G. The paper also fully characterizes all unicyclic graphs that achieve these bounds. The function f must satisfy specific requirements, which are met by several popular indices, including the Sombor index (and its reduced version), arithmetic–geometric index, sigma index, and symmetric division degree index. Consequently, the general results obtained provide bounds for several well-known indices.

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.

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.

RELATIONS BETWEEN ARITHMETIC-GEOMETRIC INDEX AND GEOMETRIC-ARITHMETIC INDEX

The arithmetic-geometric index AG(G) and the geometric-arithmetic index GA(G) of a graph G are defined as AG(G) = P uv∈E(G) dG(u)+dG(v) 2 √ dG(u)dG(v) and GA(G) = P uv∈E(G) 2 √ dG(u)dG(v) dG(u)+dG(v) , where E(G) is the edge set of G, and dG(u) and dG(v) are the degrees of vertices u and v, respectively. We study relations between AG(G) and GA(G) for graphs G of given size, minimum degree and maximum degree. We present lower and upper bounds on AG(G) + GA(G), AG(G) − GA(G) and AG(G) · GA(G). All the bounds are sharp.

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.

A Computational Approach on Fenofibrate Drug Using Degree-Based Topological Indices and M-polynomials

Fenofibrate is a drug approved by FDA Food and Drugs Administration used to treat patients with Hypertrigly ceridemia primary hypercholesterolemia or mixed dyslipidemia. It reduces low-density lipoprotein cholesterol (LDL-c), total cholesterol, triglycerides, and apolipoprotein B and increases high -density lipoprotein cholesterol (HDL-C) in adults. The sum and multiplicative versions of several topological indices such as General Zagreb, General Randic, Arithmetic Geometric Index, Inverse sum (Indeg) Index, Symmetric Division (Deg) Index, Forgotten Indices M-polynomials of Fenofibrate are computed in this article.

Topological Indices and Structural Properties of Cubic Power Graph of Dihedral Group

The cubic power graph of finite group G with identity element e, is an undirected finite, simple graph in which a pair of distinct vertices x, y have an edge iff xy = z3 or yx = z3 for any z ∈ Dn with z3 ≠ e. In this paper, we have studied the structural representation of the cubic power graph of the dihedral group and various structural properties such as clique, girth, vertex degree, chromatic number, independent number, matching number, perfect matching, dominating number, etc. We have also calculated various topological indices such as the Harary index, the first and second Zagreb indices, the Wiener and hyper-Wiener indices, the Schultz index, the harmonic index, the general Randic index, the eccentric connectivity index, the Gutman index, the atomic-bond connectivity index, and the geometricarithmetic index of the cubic power graph of dihedral group Dn when gcd(n, 3) = 1.

Prioritizing Asthma Treatment Drugs through Multicriteria Decision Making.

Asthma is a medical condition characterized by inflammation, narrowing, and swelling of a person's airways, leading to increased mucus production and difficulties in breathing. Topological indices are instrumental in assessing the physical and chemical attributes of these asthma drugs. As resistance to current treatments continues to emerge and undesirable side effects are linked to certain medications, the search for novel and enhanced drugs becomes a top priority. In this study, the examination of 19 distinct asthma medications was focused. In this study, quantitative structure-activity relationship (QSAR) and quantitative structure-property relationship (QSPR) modeling, in combination with multicriteria decision-making (MCDM) technique VIKOR (VIekriterijumsko KOmpromisno Rangiranje) were employed on asthma drugs, to achieve the most favorable rankings for each asthma drug, taking into account their distinct properties. The topological indices employed for QSPR modeling were Randic index, reciprocal Randic index, Zagreb indices, hyper-Zagreb index, harmonic index, geometric arithmetic index, and forgotten index.

The minimal chemical tree for the difference between geometric–arithmetic and Randić indices

AbstractTopological indices are numerical parameters derived from the structural information of chemical compounds. By providing a quantitative description of molecular structures, topological indices enable researchers to predict various properties and behaviors of molecules. The Randić index () and the geometric–arithmetic index () are widely recognized topological indices. It is observed that, for any given graph . We aim to investigate the gap between and for chemical tree. The complete characterization of minimal chemical tree for is carried out here. This article offers an interesting finding that, whereas and provide same minimal chemical trees, yields minimal tree structures that are totally different from and . Moreover is observed to correlate well with physico‐chemical properties of octanes.

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 THE PRODUCT OF GEOMETRIC-ARITHMETIC, SOMBOR AND FIRST ZAGREB INDICES

Assume Ω is a connected and simple graph with edge set E(Ω) and vertex set V (Ω). In chemical graph theory, the Geometric-Arithmetic index, Sombor index, and first Zagreb index of graph are three well-defined and studied topological indices. In the present study, we introduce the new graph invariant, we call it as GSM(Ω) index, this new graph invariant is the product of Geometric-Arithmetic index, Sombor index, and first Zagreb index. Furthermore we discuss the effect on GSM(Ω) of inserting and deleting an edge into a graph Ω. In addition, we investigate the connections between the GSM(G) index and various well-studied topological indices.We discussed nano JD*-Homeomorphisms in nano topological spaces in this study. We address the fundamental characteristics and noteworthy characterizations of nano JD*-Homeomorphisms in nano topological spaces. Our analysis demonstrates how they relate to other ideas already known, such as nano semihomeomorphisms, nano pre-homeomorphisms, and nano g-homeomorphisms

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.

The Geometric–Arithmetic index of trees with a given total domination number

Let G=(V,E) be a simple connected graph with vertex set V and edge set E. In Vukičević and Furtula (2009), Vukičević defined a new topological index, named geometric–arithmetic index of a graph G and denoted by GA(G), as GAG=∑uv∈E2dudvdu+dv,where du and dv denote the degrees of vertices u and v, respectively. We obtain an upper bound for the geometric–arithmetic index of trees in terms of order and the total domination number, and we characterize the extremal trees for this bound. Additionally, by a known relation between the geometric–arithmetic and arithmetic–geometric indices, we get a new lower bound for the arithmetic–geometric index.

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 ( P z ) and tetrakis porphyrazine ( T P z ) 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.

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.

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 the Distance Spectrum and Distance-Based Topological Indices of Central Vertex-Edge Join of Three Graphs

In this paper, we introduce a new graph operation based on a central graph called central vertex-edge join (denoted by $G_{n_1}^C \triangleright (G_{n_2}^V\cup G_{n_3}^E)$) and then determine the distance spectrum of $G_{n_1}^C \triangleright (G_{n_2}^V\cup G_{n_3}^E)$ in terms of the adjacency spectra of regular graphs $G_1$, $G_2$ and $G_3$ when $G_1$ is triangle-free. As a consequence of this result, we construct new families of non-D-cospectral D-equienergetic graphs. Moreover, we determine bounds for the distance spectral radius and distance energy of the central vertex-edge join of three regular graphs. In addition, we provide its results related to graph invariants like eccentric-connectivity index, connective eccentricity index, total-eccentricity index, average eccentricity index, Zagreb eccentricity indices, eccentric geometric-arithmetic index, eccentric atom-bond connectivity index, Wiener index. Using these results, we calculate the topological indices of the organic compounds Methylcyclobutane $(C_5H_{10})$ and Spirohexane $(C_6H_{10})$.

Evaluation of Various Topological Indices of Flabellum Graphs

Graph theory serves as an engaging arena for the investigation of proof methods within the field of discrete mathematics, and its findings find practical utility in numerous scientific domains. Chemical graph theory is a specialized branch of mathematics that uses graphs to represent and analyze the structure and properties of chemical compounds. Topological indices are mathematical properties of graphs that play a crucial role in chemistry. They provide a unique way to connect the structural characteristics of chemical compounds to their corresponding molecular graphs. The flabellum graph Fn(k,j) is obtained with the help of k≥2 duplicates of the cycle graph Cn with a common vertex (known as, central vertex). Then, in j of these duplicates, additional edges are added, joining the central vertex to all non-adjacent vertices. In this article, we compute different degree-based topological indices for flabellum graphs, including some well known indices, such as the Randić index, the atom bond connectivity index, the geometric–arithmetic index, and the Zagreb indices. This research provides an in-depth examination of these specific indices within the context of flabellum graphs. Moreover, the behavior of these indices is shown graphically, in terms of the parameters j,k, and n. Additionally, we have extended the concept of the first Zagreb index, to address the issue of cybercrime. This application enables us to identify criminals who exhibit higher levels of activity and engagement in multiple criminal activities when compared to their counterparts. Furthermore, we conducted a comprehensive comparative analysis of the first Zagreb index against the closeness centrality measure. This analysis sheds light on the effectiveness and relevance of the topological index in the context of cybercrime detection and network analysis.

Eccentricity-based topological indices of hexagonal networks

ABSTRACT Chemical graph theory acts as a tool for converting molecular information to a numerical parameter. Topological indices are numeric quantities that are related to the physio-chemical properties of chemical compounds. Distance- and Degree-based topological indices and Counting-related polynomials are three prominent and extensively researched classes of topological indices. In all these categories, distance-based topological descriptors have a significant impact on chemical graph theory. In this paper, the ‘Total eccentricity index ζ ( G ) , the Geometric-arithmetic index G A 4 ( G ) , Eccentricity-based Zagreb indices M 1 ∗ ( G ) , M 1 ∗ ∗ ( G ) and M 2 ∗ ( G ) , the Average eccentricity index a v e c ( G ) and the Atom-bond connectivity index A B C 5 ( G ) ’ of hexagonal networks are computed.

Degree-based topological indices of boron nanotubes

In the past two decades, boron nanotubes have received significant attention from researchers and scientists due to their wide-ranging applications in electronics, nanodevices, optical engineering, nanobiotechnology, and cosmetics. These nanotubular structures composed of boron present exceptional electrical and mechanical properties, making them highly potential nanomaterials. In this article, we study the molecular structure of significant classes of boron nanotubes, namely, trihexagonal boron nanotubes, triangular boron nanotubes, and boron-α nanotubes. Furthermore, we calculate various topological indices for these nanotubes, including the augmented Zagreb index, Sombor index, reduced Sombor index, sum-connectivity index, and arithmetic–geometric index. These indices hold substantial importance in assessing the physical, chemical, and biological characteristics of boron nanotubes.