Zagreb Indices Research Articles

Bounds on the modified second Zagreb index

Let G be a non-trivial connected graph. If the vertices vi and vj of G are adjacent, we write i∼j, otherwise we write i≁j. Denote by di the degree of the vertex vi of G. The modified second Zagreb index and coindex of G are defined as M2∗(G)=∑i∼j1didj and M2∗¯(G)=∑i≁j1didj, respectively. Several lower and upper bounds on the modified second Zagreb index/coindex are derived and all the graphs attaining these bounds are characterized. The obtained bounds can be used to deduce bounds for various other topological indices.

Switching checkerboards in (0,1)-matrices

In order to study M(R,C), the set of binary matrices with fixed row and column sums R and C, we consider submatrices of the form (1001) and (0110), called positive and negative checkerboard respectively. We define an oriented graph of matrices G(R,C) with vertex set M(R,C) and an arc from A to A′ indicates you can reach A′ by switching a negative checkerboard in A to positive. We show that G(R,C) is a directed acyclic graph and identify classes of matrices which constitute unique sinks and sources of G(R,C). Given A,A′∈M(R,C), we give necessary conditions and sufficient conditions on M=A′−A for the existence of a directed path from A to A′.We then consider the special case of M(D), the set of adjacency matrices of graphs with fixed degree distribution D. We define G(D) accordingly by switching negative checkerboards in symmetric pairs. We show that Z2, an approximation of the spectral radius λ1 based on the second Zagreb index, is non-decreasing along arcs of G(D). Also, λ1 reaches its maximum in M(D) at a sink of G(D). We provide simulation results showing that applying successive positive switches to an Erdős-Rényi graph can significantly increase λ1.

ON LEAP ZAGREB INDICES OF A SPECIAL GRAPH OBTAINED BY SEMIGROUPS

In 2013, Das et al. defined the monogenic semigroup graphs [10]. And, various topological indices of the monogenic semigroup graphs have been calculated so far [3,21]. The aim of this study is to continue to create formulas for the topological indices of these special graphs. In this study, we give exact formulae for various the leap Zagreb indices of this special algebraic graph obtained from monogenic semigroups.

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.

Entropy analysis of nickel(II) porphyrins network via curve fitting techniques

Nickel(II) porphyrins typically adopt a square planar coordination geometry, with the nickel atom located at the center of the porphyrin ring and the coordinating atoms arranged in a square plane. The additional atoms or groups coordinated to the nickel atom in nickel(II) porphyrins are called ligands. Porphyrins have been investigated as potential agents for imaging and treating cancer due to their ability to selectively bind to tumor cells and be used as sensors for a variety of analytes. Nickel(II) porphyrins are relatively stable compounds, with high thermal and chemical stability. They can be stored in a solid state or in solution without significant degradation. In this study, we compute several connectivity indices, such as general Randi’c, hyper Zagreb, and redefined Zagreb indices, based on the degrees of vertices of the chemical graph of nickel porphyrins. Then, we compute the entropy and heat of formation NiP production, among other physical parameters. Using MATLAB, we fit curves between various indices and the thermodynamic properties parameters, notably the heat of formation and entropy, using various linearity- and non-linearity-based approaches. The method’s effectiveness is evaluated using R^2, the sum of squared errors, and root mean square error. We also provide visual representations of these indexes. These mathematical frameworks might offer a mechanism to investigate the thermodynamical characteristics of NiP’s chemical structure under various circumstances, which will help us understand the connection between system dimensions and these metrics.

Topological indices and QSPR analysis of some chemical structures applied for the treatment of heart patients

AbstractIn this study, various beta‐blocker drugs used for heart disease were analyzed, and their degree‐based topological indices derived from the M‐polynomial were calculated. Linear and quadratic regression analysis was used to obtain quantitative structure‐property relationship models between the topological indices and eight the physicochemical properties of the drugs to determine their effectiveness. The results show that the harmonic index was the best predictor for boiling point, flashpoint, and enthalpy of vaporization, while the redefined third Zagreb index was effective for polarizability, molar refractivity, and molar volume. The inverse sum indeg index was found to be effective for molar refractivity, and the second modified Zagreb index was surface tension in linear regression models. In addition, the redefined third Zagreb index was the best predictor for polarizability and molar refractivity, while the second modified Zagreb index was effective for molar volume. The SDD index was found to be effective for surface tension in quadratic regression models.

On the first outdegree Zagreb index of a digraph

Let D be a digraph of order n and having a arcs. Let d1+,d2+,…,dn+ be the vertex outdegrees of D. The first Zagreb index of D is denoted by Zg+(D) and is defined as Zg+(D)=∑i=1ndi+. We obtain several upper and lower bounds for the first outdegree Zagreb index Zg+(D) of a digraph D in terms of the number of vertices n and the number of arcs a. We characterize the extremal digraphs attaining these bounds. Further, we determine the digraphs which attain the maximum, the second maximum and the third maximum value for Zg+(D) among all digraphs of order n and among all bipartite digraphs of order n. We determine the digraphs which attain the maximum value for Zg+(D) among all digraphs with its underlying graph having m≤n+2 edges. Also, we discuss the orientations of a path Pn and show that the path with n−1 vertices of outdegree 1 attains the minimum value for ZG+(D).

Retracted: Computing Correlation among the Graphs under Lexicographic Product via Zagreb Indices

Retracted: Computing Correlation among the Graphs under Lexicographic Product via Zagreb Indices

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

Let G=(V,E) be a connected graph, and d(u) the degree of vertex u∈V. We define the general Z-type index of G as Zα,β(G)=∑uv∈E[d(u)+d(v)−β]α, where α and β are two real numbers. This generalizes several famous topological indices, such as the first and second Zagreb indices, the general sum-connectivity index, the reformulated first Zagreb index, and the general Platt index, which have successful applications in QSPR/QSAR research. Hence, we are able to study these indices in a unified approach.Let C(π) the set of connected graphs with degree sequence π. In the present paper, under different conditions of α and β, we show that:(1) There exists a so-called BFS-graph having extremal Zα,β index in C(π);(2) If π is the degree sequence of a tree, a unicyclic graph, or a bicyclic graph, with minimum degree 1, then there exists a special BFS-graph with extremal Zα,β index in C(π);(3) The so-called majorization theorem of Zα,β holds for trees, unicyclic graphs, and bicyclic graphs.As applications of the above results, we determine the extremal graphs with maximum Zα,β index for α>1 and β≤2 in the set of trees, unicyclic graphs, and bicyclic graphs with given number of pendent vertices, maximum degree, independence number, matching number, and domination number, respectively. These extend the main results of some published papers.

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.

On Degree-Based Topological Indices of Toeplitz Graphs

In this research paper, the determination of the orientation of a linear interpolation in an interconnected graph can be achieved by measuring its distance from a group of sonar stations strategically positioned within the graph. The study utilizes the metric dimension of toeplitz graphs. Several indices play a crucial role in analyzing motivating activities within such complex structures. The indices covered in this study include the general connectivity index of the toeplitz graph, zagreb indices, symmetric division degree index and randic indices, among others.

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.

Atom Bound Connectivity, Zagreb, Sombor, Nirmala and Harmonic Indices of Dendrimers

A nanostar is a very small molecule ranging in size from 1-100 nanometers. It is typically composed of materials such as polymers, silica and others. Nanostars are of great significance in various fields, including drug technology, energy technology and other applications. The topological index is a numerical depiction of a molecular structure. Within chemical graphs, the atoms and the chemical bonds connecting them are symbolized by vertices and edges, respectively. Topological indices help to predict the topological properties of nanostar dendrimers, such as molecular weight and size. In this article, we computed the first Zagreb index, the second Zagreb Index, the third Zagreb index, Sombor index, ABC index, Nirmala index and Harmonic index of two new families of nanostar dendrimers such as is also known as aminoisophthalate diester monomer and known as Poly (amidoamine).

Computational Studies on Diverse Characterizations of Molecular Descriptors for Graphyne Nanoribbon Structures.

Materials made of graphyne, graphyne oxide, and graphyne quantum dots have drawn a lot of interest due to their potential uses in medicinal nanotechnology. Their remarkable physical, chemical, and mechanical qualities, which make them very desirable for a variety of prospective purposes in this area, are mostly to blame for this. In the subject of mathematical chemistry, molecular topology deals with the algebraic characterization of molecules. Molecular descriptors can examine a compound's properties and describe its molecular topology. By evaluating these indices, researchers can predict a molecule's behavior including its reactivity, solubility, and toxicity. Amidst the captivating realm of carbon allotropes, γ-graphyne has emerged as a mesmerizing tool, with exquisite attention due to its extraordinary electronic, optical, and mechanical attributes. Research into its possible applications across numerous scientific and technological fields has increased due to this motivated attention. The exploration of molecular descriptors for characterizing γ-graphyne is very attractive. As a result, it is crucial to investigate and predict γ-graphyne's molecular topology in order to comprehend its physicochemical characteristics fully. In this regard, various characterizations of γ-graphyne and zigzag γ-graphyne nanoribbons, by computing and comparing distance-degree-based topological indices, leap Zagreb indices, hyper leap Zagreb indices, leap gourava indices, and hyper leap gourava indices, are investigated.

Several Zagreb indices of power graphs of finite non-abelian groups

Molecular topology can be described by using topological indices. These are quantitative measures of the essential structural features of a proposed molecule calculated from its molecular structure. It is a numerical value obtained from a molecular configuration that reflects the significant physical characteristics of the suggested molecule. Numerous physical properties, chemical reactivity, and biological activity are correlated with the chemical composition using an algebraic number. The power graph P(G) of a finite group G is a graph whose vertex set is G and in which two distinct vertices are connected by an edge when one element is an integral power of the other. This article investigates a wide range of degree-based topological descriptors for power graphs of various finite groups. We find numerous Zagreb indices (given in Table 1) of power graphs of finite non-cyclic and cyclic groups, dihedral, and generalized quaternion groups.

ONTRANSFORMEDGRAPHS

The network systems and graphical analysis through the study of structural characteristics is a vast field of growing importance in research. Topological indices have asignificant and crucial role in the study of structure–property relationships. In this paper, we examine constructional transformed networks constructed by unique vertex-edge incidence and mutual adjacency associations. Expressions for the first and second hyper Zagreb indices and co-indices of these transformed networks and their complements are obtained.

COMPUTATION OF b-CHROMATIC TOPOLOGICAL INDICES OF SOME GRAPHS AND ITS DERIVED GRAPHS

The two fastest-growing subfields of graph theory are graph coloringand topological indices. Graph coloring is assigning the colors/values to theedges/vertices or both. A proper coloring of the graph G is assigning colors/valuesto the vertices/edges or both so that no two adjacent vertices/edges share the samecolor/value. Recently, studies involving Chromatic Topological indices that dealtwith different graph coloring were studied. In such studies, the vertex degrees getreplaced with the colors, and the computation is carried out based on the topologicalindex of our choice. We focus on b-Chromatic Zagreb indices and b-Chromaticirregularity indices in this work. This paper discusses the b-Chromatic Zagreb indicesand b-Chromatic irregularity indices of the gear graph, star graph, and itsderived graphs such as the line and middle graph.

TWO TOPOLOGICAL INDICES OF TWO NEW VARIANTS OF GRAPH PRODUCTS

Graph operations are essential to developing advanced network structures from simple graphs. In [12] they defined two new variants of corona product and discovered their topological indices. Inthis Study, we extended the work and obtain the formulas of Y-Index and Redefined third Zagreb index for corona join product and Sub-division vertex join product of graphs

Extremal hyper-Zagreb index of trees of given segments with applications to regression modeling in QSPR studies

Introduced by Gutman & Trinajstić in 1972, the two well-known Zagreb indices efficiently determine the total π-electron energy of alternant hydrocarbons. The hyper-Zagreb index is a variant of the classical first Zagreb index. A tree is an acyclic graphical structure. A segment in a tree is equivalent to a vertex of degree two. This paper derives sharp upper and lower bounds on the hyper-Zagreb index of trees with given number of segments. We also characterize the corresponding trees achieving those bounds. The second part of the paper studies the predictive potential of the hyper-Zagreb index for determining physicochemical properties such as the enthalpies of combustion, formation, sublimation, and vaporization of monocarboxylic acids. Our statistical analysis asserts that the hyper-Zagreb index correlates these physicochemical properties of monocarboxylic acids with correlation coefficient ρ>0.99. The strongest correlation by the hyper-Zagreb index was obtained with the enthalpy of combustion having correlation coefficient ρ=0.999991. This warrants the further employability of the hyper-Zagreb index in QSPR modeling. This potential applicability justifies the construction of the hyper-Zagreb graphical index whose motivation of defining was purely mathematical.