Multiplicative Zagreb Indices Research Articles

Some applications of multiplicative Zagreb index

Some applications of multiplicative Zagreb index

Predicting enthalpy of formation of benzenoid hydrocarbons and ordering molecular trees using general multiplicative Zagreb indices

Many existing studies show that there exists a strong relationship between structures and characteristics of molecules. Topological indices are often used in modeling the properties of chemical compounds and biological activities in theoretical chemistry. Topological indices are numerical values associated with structures of molecules in such a way that they remain constant under graph isomorphism. Multiplicative Zagreb indices are among the famous topological indices that have been explored by numerous researchers in the last few years. The first objective of the present paper is to examine the importance of general multiplicative Zagreb indices for forecasting the enthalpy of formation of hydrocarbons using a data set of 25 benzenoid hydrocarbons. The second objective of this paper is to study molecular trees with a given order and with a given number of branching vertices or segments using general multiplicative (first and second) Zagreb indices. Sharp lower/upper bounds on these Zagreb indices for the aforementioned molecular trees are obtained and the graphs attaining these bounds are determined. Bounds on the classical multiplicative Zagreb and Narumi-Katayama indices are corollaries of the obtained results.

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.

Connection Number-based Multiplicative Zagreb Indices of Chemical Structures

Abstract: A topological index is a quantitative measure of molecular structure and connection. It also estimates the toxicological, structural, biological, and physicochemical characteristics of a chemical molecule. The use of nano-structured graphs in the study of chemistry is very extensive. Melem, one of the most significant tri−s−triazine compounds, is a nucleophilic reagent that may be employed in reactions with derivatives of phthalic acid. Materials with borophene nanostructures are also exploited in cross-disciplinary research. In comparison to carbon hexagonal nanotubes, boron triangular nanotubes are thought to be a superior conductor. The goal of computing the multiplicative Zagreb connection indices for Chemical graphs is to understand the complexity, connectedness, and chemical activity of molecules. Information is useful in a number of applications, including drug design, chemical synthesis, and property prediction. In this paper, we compute the first, second, third, fourth, and fifth multiplicative Zagreb connection indices (ZCIs) of the melem chain MC(s), the borophene chain B36(s), and the boron triangular sheet BTS(m,s).

Analysis of the Zagreb Indices over the Weakly Zero-Divisor Graph of the Ring Zp×Zt×Zs

Let R be a commutative ring with identity, and Z(R) be the set of zero-divisors of R. The weakly zero-divisor graph of R denoted by WΓ(R) is an undirected (simple) graph with vertex set Z(R)*, and two distinct vertices x and y are adjacent, if and only if there exist r∈ann(x) and s∈ann(y), such that rs=0. Importantly, it is worth noting that WΓ(R) contains the zero-divisor graph Γ(R) as a subgraph. It is known that graph theory applications play crucial roles in different areas one of which is chemical graph theory that deals with the applications of graph theory to solve molecular problems. Analyzing Zagreb indices in chemical graph theory provides numerical descriptors for molecular structures, aiding in property prediction and drug design. These indices find applications in QSAR modeling and chemical informatics, contributing to efficient compound screening and optimization. They are essential tools for advancing pharmaceutical and material science research. This research article focuses on the basic properties of the weakly zero-divisor graph of the ring Zp×Zt×Zs, denoted by WΓ(Zp×Zt×Zs), where p, t, and s are prime numbers that may not necessarily be distinct and greater than 2. Moreover, this study includes the examination of various indices and coindices such as the first and second Zagreb indices and coindices, as well as the first and second multiplicative Zagreb indices and coindices of WΓ(Zp×Zt×Zs).

Topological Evaluation of Four Para-Line Graphs Absolute Pentacene Graphs Using Topological Indices

A real-number to molecular structure mapping is a topological index. It is a graph invariant method for describing physico-chemical properties of molecular structures specific substances. In that article, We examined pentacene’s chemical composition. The research on the subsequent indices is reflected in our paper, we conducted an analysis of several indices including general randic connectivity index, first general zagreb index, general sum-connectivity index, atomic bond connectivity index, geometric-arithmetic index, fifth class of geometric-arithmetic indices, hyper-zagreb index, first and second multiple zagreb indices for a four para-lines graphs of linear [n]-pentacene and multi-pentacene.

Relations Between the Energy and Topological Indices of a Graph

In this paper, we give various lower and upper bounds for the energy of graphs in terms of several topological indices of graphs: the first general multiplicative Zagreb index, the general Randić index, the general zeroth-order Randić index, the redefined Zagreb indices, and the atom-bond connectivity index. Moreover, we obtain new bounds for the energy in terms of certain graph invariants as diameter, girth, algebraic connectivity and radius.

A Unified Approach for Extremal General Exponential Multiplicative Zagreb Indices

The study of the maximum and minimal characteristics of graphs is the focus of the significant field of mathematics known as extreme graph theory. Finding the biggest or smallest graphs that meet specified criteria is the main goal of this discipline. There are several applications of extremal graph theory in various fields, including computer science, physics, and chemistry. Some of the important applications include: Computer networking, social networking, chemistry and physics as well. Recently, in 2021 exponential multiplicative Zagreb indices were introduced. In generalization, we introduce the generalized form of exponential multiplicative Zagreb indices for α∈R+\{1}. Furthermore, to see the behaviour of generalized first and second exponential Zagreb indices for α∈R+\{1}, we used a transformation method. In term of the two newly developed generalized exponential multiplicative Zagreb indices, we will investigate the extremal bicyclic, uni-cyclic and trees graphs. Four graph transformations are used and some bounds are presented in terms of generalized exponential multiplicative Zagreb indices.

On multiplicative universal Zagreb and its subsequent indices of C4C8 carbon nanostructures

Mathematical chemistry studies the chemical structure of molecules. Topological indices are numerical values which associates the chemical structure with the physical and chemical properties. Multiplicative Universal Zagreb indices are generalized degree-based topological indices which gave rise to several indices like first and second multiplicative, Zagreb and hyper-Zagreb indices, multiplicative sum and product conductivity indices. This research is designed to study the first and second multiplicative Universal Zagreb indices of carbon nanostructures (carbon nanosheet, nanotube and nanotorus) of the same chemical formula [Formula: see text] We also depict the values of first and second multiplicative, Zagreb and hyper-Zagreb indices as well as multiplicative sum and product connectivity indices for the mentioned structures. The graphical comparison for each of the multiplicative Zagreb indices is presented for all the carbon nanostructures with the same chemical formula [Formula: see text]

Mathematical aspects and topological properties of two chemical networks

<abstract><p>Graphs give a mathematical model of molecules, and thery are used extensively in chemical investigation. Strategically selections of graph invariants (formerly called "topological indices" or "molecular descriptors") are used in the mathematical modeling of the physio-chemical, pharmacologic, toxicological, and other aspects of chemical compounds. This paper describes a new technique to compute topological indices of two types of chemical networks. Our research examines the mathematical characteristics of molecular descriptors, particularly those that depend on graph degrees. We derive a compact mathematical analysis and neighborhood multiplicative topological indices for product of graphs ($ \mathcal{L} $) and tetrahedral diamond lattices ($ \Omega $). In this paper, the fifth multiplicative Zagreb index, the general fifth multiplicative Zagreb index, the fifth multiplicative hyper-Zagreb index, the fifth multiplicative product connectivity index, the fifth multiplicative sum connectivity index, the fifth multiplicative geometric-arithmetic index, the fifth multiplicative harmonic index and the fifth multiplicative redefined Zagreb index are determined. The comparison study of these topological indices is also discussed.</p></abstract>

Extremal quasi-unicyclic graphs with respect to the general multiplicative Zagreb indices

The general multiplicative Zagreb indices are modified versions of the famous Zagreb indices, which can help to predict the various activities and properties of organic compounds. In this article, the minimum and maximum general multiplicative Zagreb indices of quasi-unicyclic graphs with given order, with fixed number of pendant vertices and with perfect matchings are investigated. Furthermore, the corresponding extremal quasi-unicyclic graphs are identified.

Computing Topological Invariants of Deep Neural Networks.

A deep neural network has multiple layers to learn more complex patterns and is built to simulate the activity of the human brain. Currently, it provides the best solutions to many problems in image recognition, speech recognition, and natural language processing. The present study deals with the topological properties of deep neural networks. The topological index is a numeric quantity associated to the connectivity of the network and is correlated to the efficiency and accuracy of the output of the network. Different degree-related topological indices such as Zagreb index, Randic index, atom-bond connectivity index, geometric-arithmetic index, forgotten index, multiple Zagreb indices, and hyper-Zagreb index of deep neural network with a finite number of hidden layers are computed in this study.

MULTIPLICATIVE ZAGREB INDICES OF FOUR NEW F-SUMS OF GRAPHS

For molecular graph G, the first multiplicative Zagreb index is defined as the product of squares of degree of all vertices of graph and the second multiplicative Zagreb index is defined as Q 2 (G) = Q u∈V (G) dG(u) dG(u) . In this paper, we obtain first and second multiplicative Zagreb indices of four new F-Sums of graphs.

On maximum Zagreb indices of bipartite graphs with a given connectivity

For a graph, the first (multiplicative) Zagreb index is equal to the sum (product) of squares of the vertex degrees, and the second (multiplicative) Zagreb index is equal to the sum (product) of products of the degrees of a pair of adjacent vertices. In this work, by a unified approach, we determine the extremal values of these Zagreb indices in terms of the (edge) connectivity and characterize the corresponding extremal graphs among all connected bipartite graphs of order [Formula: see text]. Our results show that the extremal graphs of given (edge) connectivity regarding the Zagreb indices and multiplicative Zagreb indices do not completely coincide with other topological indices.

Computing topological indices of certain windmill graphs

Let V (G), be the vertex set of graph G, we define Vd = {v ∈ V (G) : SG(u) = d} in which SG(u) = ∑v∈NG(u) dG(v) and NG(u) = {v ∈ V(G) : uv ∈ E(G)}. In this paper we express the explicit formula for Sanskruti index S(G), fth M1 and M2 Zagreb indices, fifth M1 and M2 multiplicative Zagreb indices for Dutch windmill graph, Kulli cyclic windmill graph, Kulli path windmill graph and French windmill graph.

Minimizing vertex-degree function index for k-generalized quasi-unicyclic graphs

In this paper the problem of minimizing vertex-degree function index Hf(G) for k-generalized quasi-unicyclic graphs of order n is solved for k ≥ 1 and n ≥ 2k + 2 if the function f is strictly increasing and strictly convex. These conditions are fulfilled by general first Zagreb index 0Rα(G) if α > 1, second multiplicative Zagreb index ∏2(G) and sum lordeg index SL(G). The extremal graph is unique for k = 1, n = 4 and for k ≥ 2 and it consists from a path x1, x2, …, xn − 1 and a new vertex xn adjacent with xk, xk + 1 and xk + 2.

Extremal (n,m)-Graphs w.r.t General Multiplicative Zagreb Indices.

A topological index of a molecular graph is the numeric quantity which can predict certain physical and chemical properties of the corresponding molecule. Xu et al. introduced some graph transformations which increase or decrease the first and second multiplicative Zagreb indices and proposed a unified approach to characterize extremal (n, m)- graphs. Graph transformations are used to find the extremal graphs, these transformations either increase or decrease the general multiplicative Zagreb indices. By applying the transformations which increase the general multiplicative Zagreb indices we find the graphs with maximal general multiplicative Zagreb indices and for minimal general Zagreb indices we use the transformations which decrease the index. In this paper, we extend the Xu's results and show that the same graph transformations increase or decrease the first and second general multiplicative Zagreb indices for . As an application, the extremal acyclic, unicyclic and bicyclic graphs are presented for general multiplicative Zagreb indices. By applying the transformation we investigated that in the class of acyclic, unicyclic and bicyclic graphs, which graph gives the minimum and the maximum general multiplicative Zagreb indices.

Topological Indices of Families of Bistar and Corona Product of Graphs

Topological indices are graph invariants that are used to correlate the physicochemical properties of a chemical compound with its (molecular) graph. In this study, we study certain degree‐based topological indices such as Randić index, Zagreb indices, multiplicative Zagreb indices, Narumi–Katayama index, atom‐bond connectivity index, augmented Zagreb index, geometric‐arithmetic index, harmonic index, and sum‐connectivity index for the bistar graphs and the corona product , where represents the complement of complete graph Kn.

Retracted] On Multiplicative Topological Invariants of Magnesium Iodide Structure

In recent times, the applications of graph theory in molecular and chemical structure research have far exceeded human expectations and have grown exponentially. In this paper, we have determined the multiplicative Zagreb indices, multiplicative hyper‐Zagreb indices, multiplicative universal Zagreb indices, sum and product connectivity of multiplicative indices, multiplicative atom‐bond connectivity index, and multiplicative geometric‐arithmetic index of a famous crystalline structure, magnesium iodide (MgI2).

On Some Neighbourhood Degree-Based Multiplicative Topological Indices and Their Applications

In this paper, we mainly concern with some neighborhood degree-based multiplicative topological indices that are introduced by some eminent researchers in the past three years. According to the suggestion of the “International Academy of Mathematical Chemistry,” we tested their predictive ability of physicochemical properties of molecules with the help of a data set of octane isomers. Here, we also modified the general fifth multiplicative Zagreb index as a new one and computed this index for some line graphs of subdivision graphs of some union graphs of hexagons. Hence, computed some other multiplicative topological indices with the help of this index.