Augmented Zagreb Index Research Articles

Topology of Edge Contracted Möbius Ladder: Indices and Dimension

Assigning numbers to graphs serves as a fundamental tool in graph theory and enables various analytical, computational, and visualization techniques for studying and understanding graph structures and properties. Edge Contraction is a process that separates edges from the graph while also joining the two previously linked vertices. In this paper, we study metric dimension and degree-based topological indices of Möbius Ladder and edge contracted Möbius Ladder like first Zagreb index, second Zagreb index, first Zagreb Hyper index, second Zagreb Hyper index, Augmented Zagreb index, Randic index, general Randic index, and Harmonic index.

Complete solution to open problems on exponential augmented Zagreb index of chemical trees

One of the crucial problems in combinatorics and graph theory is characterizing extremal structures with respect to graph invariants from the family of chemical trees. Cruz et al. (2020) [7] presented a unified approach to identify extremal chemical trees for degree-based graph invariants in terms of graph order. The exponential augmented Zagreb index (EAZ) is a well-established graph invariant formulated for a graph G asEAZ(G)=∑vivj∈E(G)e(didjdi+dj−2)3, where di signifies the degree of vertex vi, and E(G) is the edge set. Due to some special counting features of EAZ, it was not covered by the aforementioned unified approach. As a result, the exploration of extremal chemical trees for this invariant was posed as an open problem in the same article. The present work focuses on generating a complete solution to this problem. Our findings offer maximal and minimal chemical trees of EAZ in terms of the graph order n.

Open problem on the maximum exponential augmented Zagreb index of unicyclic graphs

Open problem on the maximum exponential augmented Zagreb index of unicyclic graphs

Extremal properties of connected 4-cyclic graphs of a topological index related to chemical graphs.

Augmented Zagreb index (AZI) is an important vertex-degree-based topological index with many applications especially in chemistry. In this article, minimum value of is obtained and the corresponding extremal graph is characterized in the class of connected 4-cyclic graphs with seven or more vertices. These results can be used to characterize numerous chemical properties of chemical compounds having 4-cyclic structures.

Topological Insights into Nanostar Dendrimers by Computing the Augmented Zagreb Index.

The field of nanobiotechnology uses precise nanofabrication techniques to advance our understanding and control of biological systems. Due to their remarkable properties, dendrimers, which are hyperbranched macromolecular structures with distinct and well-defined architectures, have emerged as pivotal entities within this field. They are gaining increasing attention for their potential to catalyze a paradigm shift in medical therapeutics, biotechnological applications, and advanced material sciences. This paper focuses on a novel analytical expression and determines the precise value of the augmented Zagreb index, a topological descriptor, for eight classes of nanostar dendrimers. The Zagreb index is a topological invariant to predict molecular behaviour and reactivity. In this paper, we have explored its application in characterizing the branching of nanostar dendrimers through computational modelling and mathematical rigor. Our research has measured the augmented Zagreb index for nanostar dendrimers, which fall into eight distinct classes. The results better explain the relationship between the dendrimers' topology and chemical properties. This correlation has implications for their structural stability and reactivity, potentially leading to new applications. Developing the augmented Zagreb index for nanostar dendrimers is a significant breakthrough in nanobiotechnology. Based on the correlation between the calculated topological index and the corresponding molecular attributes, our analytical approach has opened up new possibilities for designing and synthesizing dendrimers tailored to specific functions in medical and material science applications. This precise topological quantification could significantly enhance the utility and functionalization of dendrimers in cutting-edge nanotechnological applications.

Augmented Zagreb Index of Corona Product of Some Special Graphs

Abstract: Let GV, E be a simple undirected graph. The Augmented Zagreb Index of a graph G is defined as     3 2            uv E G du dv du dv AZI G , Where du is the degree of the vertex u in G . In this paper, the exact expression for Augmented Zagreb Index of different product of graphs like Comb, Wheel, Fan,andSun graph.

Computation of reverse neighbourhood degree-based topological indices for the transition metal phthalocyanine polymers (poly-TMPc)

In this article, we discuss the reverse neighbourhood degree-based topological indices for the transition metal phthalocyanine polymers (poly- TMPc). Degree-based topological indices have been extensively studied and linked to a variety of chemical properties. An even more recently established methodology for assessing chemical systems and geometries is the use of numerical descriptors based on the degree of definition defined by the reverse-degree concept technique. It highlights molecular characteristics as numerical descriptors based on the reverse-degree concept’s degree of definition and delivers numerical descriptors in algebraic form. On the other hand, due to their high catalytic activity, photostability, resistance to severe environments, and adaptable properties, transition metal phthalocyanines are a fascinating class of organic semiconductors. Furthermore, different types of reverse neighbourhood topological indices, such as the augmented Zagreb index, first and second Zagreb indexes, and Randi ć indexes, are defined and used to assess the complexity, stability, and other properties of poly-TMPc(m,n).

On the exponential augmented Zagreb index of graphs

On the exponential augmented Zagreb index of graphs

On chemical trees with minimum augmented Zagreb index

On chemical trees with minimum augmented Zagreb index

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.

Complete Characterization of Chemical Trees with Maximal Augmented Zagreb Index

Complete Characterization of Chemical Trees with Maximal Augmented Zagreb Index

The k-apex trees with minimum augmented Zagreb index

For a connected graph G on at least three vertices, the augmented Zagreb index (AZI) of G is defined asAZI(G)=∑uv∈E(G)(d(u)d(v)d(u)+d(v)−2)3, being a topological index well-correlated with the formation heat of alkanes. A k-apex tree G is a connected graph admitting a k-subset X⊂V(G) such that G−X is a tree, while G−S is not a tree for any S⊂V(G) of cardinality less than k. By investigating some structural properties of k-apex trees, we identify the graphs minimizing the AZI among all k-apex trees on n vertices for k≥4 and n≥3(k+1). The latter solves an open problem posed in Cheng et al. (2021) [5].

Valency-Based Indices for Some Succinct Drugs by Using M-Polynomial

A topological index, which is a number, is connected to a graph. It is often used in chemometrics, biomedicine, and bioinformatics to anticipate various physicochemical properties and biological activities of compounds. The purpose of this article is to encourage original research focused on topological graph indices for the drugs azacitidine, decitabine, and guadecitabine as well as an investigation of the genesis of symmetry in actual networks. Symmetry is a universal phenomenon that applies nature’s conservation rules to complicated systems. Although symmetry is a ubiquitous structural characteristic of complex networks, it has only been seldom examined in real-world networks. The M¯-polynomial, one of these polynomials, is used to create a number of degree-based topological coindices. Patients with higher-risk myelodysplastic syndromes, chronic myelomonocytic leukemia, and acute myeloid leukemia who are not candidates for intense regimens, such as induction chemotherapy, are treated with these hypomethylating drugs. Examples of these drugs are decitabine (5-aza-20-deoxycytidine), guadecitabine, and azacitidine. The M¯-polynomial is used in this study to construct a variety of coindices for the three brief medicines that are suggested. New cancer therapies could be developed using indice knowledge, specifically the first Zagreb index, second Zagreb index, F-index, reformulated Zagreb index, modified Zagreb, symmetric division index, inverse sum index, harmonic index, and augmented Zagreb index for the drugs azacitidine, decitabine, and guadecitabine.

Study the Behavior of Drug Structures via Chemical Invariants Using TOPSIS and SAW.

Every year, various experiments emerge in which a strong link between topological chemical structures and their properties is found. These properties are numerous such as melting point, boiling point, and drug toxicity. Topological index is the functional tool to determine these properties. This research paper will analyze some of the molecular drug structures, i.e., hyaluronic acid-paclitaxel conjugates G n , anticancer drug SP[n], polyomino chain of n-cycle Z n , triangular benzenoid T n , and circumcoronene benzenoid series H k using multicriteria decision-making techniques including TOPSIS and SAW. The topological indices used in this research paper include the Randić index for α = 1, -1, 1/2, the augmented Zagreb index and the forgotten topological index.

Algebraic Properties for Molecular Structure of燤agnesium營odide

As an inorganic chemical, magnesium iodide has a significant crystalline structure. It is a complex and multi-functional substance that has the potential to be used in a wide range of medical advancements. Molecular graph theory, on the other hand, provides a sufficient and cost-effective method of investigating chemical structures and networks. M-polynomial is a relatively new method for studying chemical networks and structures in molecular graph theory. It displays numerical descriptors in algebraic form and highlights molecular features in the form of a polynomial function. We present a polynomials display of magnesium iodide structure and calculate several M-polynomials in this paper, particularly the M-polynomials of the augmented Zagreb index, inverse sum index, hyper Zagreb index and for the symmetric division index.

Computation of Degree-Based Numerical Descriptors of Porous Graphene

Graph theory is attracting much attention due to the use of devices like topological indices. A topological graph index is defined according to a certain rule. After defining a new topological index, it must be checked for a possible correlation with the properties of a particular chemical substance along with a few mathematical properties. As most of the indices involve some data related to the degrees of the vertices, types of edges, and various details of the chemical compound under the study, it is very useful in chemistry applications. The applications include drug design, modeling of a compound, the study of structure relationships, etc. In this work, the redefined Zagreb indices, ABC index, GA index, Augmented Zagreb index, neighborhood version of redefined Zagreb indices, ABC4 index, GA5 index, and Sanskruti index is computed for the chemical compound called porous graphene. The work is concluded with a detailed conclusion of the study considered.

On general [formula omitted]-type index of connected graphs

For a simple connected graph G=(V,E), define ABCα(G;β)=∑uv∈EdG(u)+dG(v)−βdG(u)dG(v)α,for any α∈R∖{0} and β∈R. This topological index generalizes the famous ABC index, augmented Zagreb index, and inverse sum indeg index. Let Γ(π) be the class of connected graphs with given degree sequence π. In this paper, under different conditions of α and β, we show that (i) For any connected graph G with uv∉E(G), ABCα(G+uv;β)>ABCα(G;β) holds; (ii) For any given degree sequence π with minimum degree 1, there exists a special extremalBFS-graph (see Definition 1.2) for ABCα(G;β) in the class of Γ(π); (iii) For any given c-cyclic degree sequence π with minimum degree 1 and c∈{0,1,2}, there exists a precise maximum extremalBFS-graph (see Definition 1.3) for ABCα(G;β) in the class of Γ(π). These extend the main results of (Chen et al., 2021; Lin et al., 2013; Lin et al., 2018; Chen and Hao, 2018).

Topological Properties of Concealed Non-Kekulean Benzenoid Hydrocarbon

A topological descriptor is regarded as a numerical parameter derived by using mathematical tools from the molecular graph of different chemical structures. The chemical graph theory is a paramount section of mathematical chemistry. In this section, there are many topological parameters, that have very useful characteristics to study the molecular structure of chemicals. In this article, we investigate topological indices of concealed non-kekulean benzenoid hydrocarbon graph. We will compute general Randic index, general sum connectivity index, sum connectivity index, general version of harmonic index, harmonic index, first, second and third Zagreb indices, SK, SK 1, SK 2 indices, augmented Zagreb index, atomic bond connectivity index, geometric arithmetic index, first and second gourava indices, first and second hyper-gourava indices and 4th version of atomic bond connectivity, 5th version of geometric arithmetic index, and the Sanskurti index.

Topological Invariants for the General Structure of Grape Seed Proanthocyanidins

In this paper, we formulate the degree-based topological indices such as first Zagreb index, second Zagreb index, third Zagreb index, atom bond connectivity index, geometric-arithmetic (GA) index, general sum connectivity index, hyper-Zagreb index, augmented Zagreb index, sum connectivity index and general Randic index Rλ (℘), for λ = © −1, 1, − 1 2 , 1 2 ª for grape seed Proanthocyanidins network. By using above-mentioned topological indices, this study provides us different outcomes.

-polynomial and deficient topological indices of identified graphs

To study the properties such as physical and chemical of compounds, the topological indices are introduced in chemical graph theory. These indices provide qualitative structure activity relationship (QSAR). Degree based topological indices are commonly used invariant in chemical graph theory. However, in this article, a new degree of vertices is introduced, called “deficiency degree”. Further, we have computed five topological indices based on the deficiency degree like “deficient first Zagreb index, deficient generalized Randić index, deficient harmonic index, deficient inverse sum index, deficient augmented Zagreb index” for identified graphs using the M -polynomial of graph.