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AbstractMolecular 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. [ABSTRACT FROM AUTHOR]

Currently topological indices have been recognized as the important areas of the graph theory and offering various applications. They are numerical quantities or measures interconnected with a graph layout or structure. These indices grant insights into multiple aspects of the graphs’s connectivity, distance and other structural properties. Properties of the chemical compounds and topological indices are correlated.Here in this paper we depict a new graphical structure Kn − dome, and assess their topological indices namely Zagreb indices in different forms, randic and somber index. Furthermore we compute the correlation coefficients between them. [ABSTRACT FROM AUTHOR]

A topological index is a numerical quantity for the structure graph of the molecule and it can be represented through graph theory. Also, its application not only in the field of chemistry can also be applied in areas including computer science, networking, etc. Hence, this paper introduces the Randic index of a vague graph and vague subgraph with their properties. The upper and lower boundaries of the Randic index of vague graphs are studied with some isomorphic properties. Likewise, the Randic index of directed vague graphs is introduced. Finally, an application of the Randic index in construction has been presented. [ABSTRACT FROM AUTHOR]

Cyclooctane is classified as a cycloalkane, characterized by the chemical formula C8H16. It consists of a closed ring structure composed of eight carbon atoms and sixteen hydrogen atoms. A cyclooctane chain typically refers to a series of cyclooctane molecules linked together. Cyclooctane and its derivatives find various applications in chemistry, materials science, and industry. Topological indices are numerical values associated with the molecular graph of a chemical compound, predicting certain physical or chemical properties. In this study, we calculated the expected values of degree-based and neighborhood degree- based topological descriptors for random cyclooctane chains. A comparison of these topological indices' expected values is presented at the end. [ABSTRACT FROM AUTHOR]

Fuzzy graphs (FGs) are efficient mathematical models for analyzing many problems of daily life. Various problems in the fields of physics, telecommunication science, psychology and genetics can be modeled and solved with the help of fuzzy graph theory. One of the most widely used types of FG is vague graph (VG). VGs play an important role in various fields such as computer science, psychology, medicine, and political sciences and are used to find effective people in an organization or social institution. Today, connectivity indices are used in computer networks and economics to find the most suitable place for construction. These indices also have many applications in medical sciences, including ultrasound and radiology. Hence, in this research, we introduce connectivity index (CI), average connectivity index (ACI), and randic index (RI) in VGs with some examples. Likewise, some new vertices such as enhancing vertex (EV), reducing vertex (RV), and neutral vertex (NV) are described. Finally, an application of CI in the selecting town for building a school is given. [ABSTRACT FROM AUTHOR]

For a k -uniform hypergraph H , we introduce degree-based indices such as the general sum-connectivity index χ a (H) = ∑ v 1 v 2 ⋯ v k ∈ E (H) [ d (v 1) + d (v 2) + ⋯ + d (v k) ] a and the general Randić index R a (H) = ∑ v 1 v 2 ⋯ v k ∈ E (H) [ d (v 1) d (v 2) ⋯ d (v k) ] a , where a ∈ ℝ , E (H) is the set of hyperedges of H and d (v i) is the degree of a vertex v i in H ; k ≥ 2 and i = 1 , 2 , ... , k. Other indices such as the first and second Zagreb index, first and second hyper-Zagreb index, classical sum-connectivity index, classical Randić index and harmonic index of a hypergraph H are special cases of the general indices. For a > 0 , we obtain upper bounds on χ a (H) and R a (H) for a uniform hypergraph H with given order, order and number of isolated vertices, order and maximum degree, order and diameter at least 2 , and lower bounds for uniform hypergraphs with given order and no isolated vertices, order and minimum degree, and order and maximum possible degree. We also present extremal graphs for all the bounds. Bounds on Zagreb indices follow from our results on general indices. [ABSTRACT FROM AUTHOR]

Summary statistics play an important role in network data analysis. They can provide us with meaningful insight into the structure of a network. The Randić index is one of the most popular network statistics that has been widely used for quantifying information of biological networks, chemical networks, pharmacologic networks, etc. A topic of current interest is to find bounds or limits of the Randić index and its variants. A number of bounds of the indices are available in literature. Recently, there are several attempts to study the limits of the indices in the Erdős–Rényi random graph by simulation. In this paper, we shall derive the limits of the Randić index and its variants of an inhomogeneous Erdős–Rényi random graph. Our results charaterize how network heterogeneity affects the indices and provide new insights about the Randić index and its variants. Finally we apply the indices to several real-world networks. [ABSTRACT FROM AUTHOR]

Connectivity measures real energization state of manifold applications and user interference in many criteria-based network systems. Randic index is a parameter of this type that can be used to measure the combined power of molecular graph or graphical network system. The concept of Randic index has been applied in a connected system in Indonesia tourism sectors under some uncertain conditions. First, Randic index of fuzzy graph (FG) and fuzzy subgraph are introduced and investigated their properties. Second, different upper and lower bounds of Randic index and their isomorphic properties in fuzzy graphs are exhibited. Third, the Randic index in directed fuzzy graph (FDG) is introduced. Due to presence of similar contribution of vertices, many formulas for calculating Randic index of regular fuzzy graphs are presented. Fourth, some similarity and distinction of Randic index with connectivity index (CI) and Wiener index (WI) in fuzzy graph are analyzed. Fifth, an algorithm and a flowchart are proposed to determine Randic index in a fuzzy graph. Finally, an application is depicted to mention the order of important crossroads and stoppages between Balai Kemambang (BKB) and Bukit Kendalisada (BKS) in Indonesia tourism sectors. [ABSTRACT FROM AUTHOR]

For an integer $ h\geq 0 $, the $ h $th order Randić index for a simple graph $ G $ is defined as $ R^{h}(G) = \sum_{\pi}\frac{1}{\sqrt{v_1(\pi)v_2(\pi)\cdots v_{h+1}(\pi)}} $, where $ \pi $ extends over all paths of length $ h $ in $ G $ and $ v_i(\pi) $ denotes the degree of the $ i $-th vertex of the path $ \pi $. In this paper, we showed that the $ h $th order Randić index $ R^{h}(T) $ of a double starlike tree $ T $ (a tree with two vertices of degrees $ m_1, m_2 > 2 $) is completely determined by its branches of length $ \leq h $. As a consequence, we proved that the double starlike trees with equal $ h $-Randić index are isomorphic, except for some special values for $ m_1 $ and $ m_2 $.

Topological 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 (R$$ R $$) and the geometric–arithmetic index (GA$$ GA $$) are widely recognized topological indices. It is observed that, GA(G)>R(G)$$ GA(G)>R(G) $$ for any given graph G$$ G $$. We aim to investigate the gap between GA$$ GA $$ and R$$ R $$ for chemical tree. The complete characterization of minimal chemical tree for GA−R$$ GA-R $$ is carried out here. This article offers an interesting finding that, whereas GA$$ GA $$ and R$$ R $$ provide same minimal chemical trees, GA−R$$ GA-R $$ yields minimal tree structures that are totally different from GA$$ GA $$ and R$$ R $$. Moreover GA−R$$ GA-R $$ is observed to correlate well with physico‐chemical properties of octanes. [ABSTRACT FROM AUTHOR]

In this work, we determine the maximum general Randić index (a general symmetric function of vertex degrees) for η 0 ≤ η < 0 among all n-vertex unicyclic graphs with a fixed maximum degree Δ and the maximum and the second maximum general Randić index for η 0 ≤ η < 0 among all n-vertex unicyclic graphs, where η 0 ≈ − 0.21 . We establish sharp inequalities and identify the graphs attaining the inequalities. Thereby, extremal graphs are obtained for the general Randić index, and certain open gaps in the theory of extremal unicyclic graphs are filled (some open problems are provided). We use computational software to calculate the Randić index for the chemical trees up to order 7 and use the statistical (linear regression) analysis to discuss the various applications of the Randić index with the physical properties of drugs on the said chemical trees. We show that the Randić index is better correlated with the heat of vaporization for these alkanes. [ABSTRACT FROM AUTHOR]

Recently, Gutman et al., defined three vertex-degree-based topological invarients and obtained mathematical properties with their applications to physico-chemical properties of alkanes. In this paper, the expressions for the Reciprocal Randíc index, Reduced Zagreb index and Reduced reciprocal Randíc index of the generalized transformation graphs Gxy and their complement graphs are obtained. [ABSTRACT FROM AUTHOR]

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In 1997, Sierpinski graphs, S (n , k) , were obtained by Klavzar and Milutinovic. The graph S (1 , k) represents the complete graph K k and S (n , 3) is known as the graph of the Tower of Hanoi. Through generalizing the notion of a Sierpinski graph, a graph named a generalized Sierpinski graph, denoted by S i e (Λ , t) , already exists in the literature. For every graph, numerous polynomials are being studied, such as chromatic polynomials, matching polynomials, independence polynomials, and the M-polynomial. For every polynomial there is an underlying geometrical object which extracts everything that is hidden in a polynomial of a common framework. Now, we describe the steps by which we complete our task. In the first step, we generate an M-polynomial for a generalized Sierpinski graph S i e (Λ , t) . In the second step, we extract some degree-based indices of a generalized Sierpinski graph S i e (Λ , t) using the M-polynomial generated in step 1. In step 3, we generate the entropy of a generalized Sierpinski graph S i e (Λ , t) by using the Randić index. [ABSTRACT FROM AUTHOR]

The Randić index of a graph G is the sum of (d G (u) d G (v)) − 1 2 over all edges u v of G, where d G (u) denotes the degree of vertex u in G. In this paper, we investigate a few graph transformations that decrease the Randić index of a graph. By applying those transformations, we determine the minimum Randić index on tricyclic graphs and characterize the corresponding extremal graphs. [ABSTRACT FROM AUTHOR]

The concept of Randić index has been extended recently for a digraph. We prove that 2 R (G) ≤ E (G) ≤ 2 Δ (G) R (G) , where G is a digraph, and R (G) denotes the Randić index, E (G) denotes the Nikiforov energy and Δ (G) denotes the maximum degree of G. Moreover, in both inequalities, we describe the graphs for which the equality holds. We also develop the concept of inner and outer energies of a vertex for directed graphs extending important inequalities known for the energy of a vertex for graphs. [ABSTRACT FROM AUTHOR]

Cyclooctane is classified as a cycloalkane, characterized by the chemical formula C8H16. It consists of a closed ring structure composed of eight carbon atoms and sixteen hydrogen atoms. A cyclooctane chain typically refers to a series of cyclooctane molecules linked together. Cyclooctane and its derivatives find various applications in chemistry, materials science, and industry. Topological indices are numerical values associated with the molecular graph of a chemical compound, predicting certain physical or chemical properties. In this study, we calculated the expected values of degree-based and neighborhood degree-based topological descriptors for random cyclooctane chains. A comparison of these topological indices’ expected values is presented at the end.

For a real number α\alpha , the general Randić index of a graph GG, denoted by Rα(G){R}_{\alpha }\left(G), is defined as the sum of (d(u)d(v))α{\left(d\left(u)d\left(v))}^{\alpha } for all edges uvuv of GG, where d(u)d\left(u) denotes the degree of a vertex uu in GG. In particular, R−12(G){R}_{-\tfrac{1}{2}}\left(G) is the ordinary Randić index, and is simply denoted by R(G)R\left(G). Let α\alpha be a real number. In this article, we show that (1)if α≥0\alpha \ge 0, Rα(L(G))≥2Rα(G){R}_{\alpha }\left(L\left(G))\ge 2{R}_{\alpha }\left(G) for any graph GG with δ(G)≥3\delta \left(G)\ge 3;(2)if α≥0\alpha \ge 0, Rα(L(G))≥Rα(G){R}_{\alpha }\left(L\left(G))\ge {R}_{\alpha }\left(G) for any connected graph GG which is not isomorphic to Pn{P}_{n};(3)if α

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 F n (k , j) is obtained with the help of k ≥ 2 duplicates of the cycle graph C n 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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

A topological index is a type of molecular descriptor that is calculated based on the molecular graph of a chemical compound. Topological indices are used for example in the development of QSAR QSPR in which the biological activity or other properties of molecules are correlated with their chemical structure. In this paper, we establish several inequalities among the molecular descriptors such as the generalized version of the first Zagreb index, the Randić index, the ABC index, AZI index, and the redefined first, second and third Zagreb indices. [ABSTRACT FROM AUTHOR]

We investigate the index If (G) = vw2E(G) f(dG(v); dG(w)) of a graph G, where f is a symmetric function of two variables satisfying certain conditions, E(G) is the edge set of G, and dG(v) and dG(w) are the degrees of vertices v and w in G, respectively. Those conditions are satisfied by functions that can be used to define the general sum-connectivity index Xa, general Randic index Ra, general reduced second Zagreb index GRMa for some a 2 R, general Sombor index SOa;b, general augmented Zagreb index AZIa;b and by one other generalization Ma;b for some a; b 2 R. The general augmented Zagreb index is a new index defined in this paper. We obtain a sharp upper bound on If for graphs with given order and connectivity, and a sharp lower bound on If for 2-connected graphs with given order. Our upper bound holds for Ma;b and SOa;b where a; b ≥ 1; Xa and Ra where a ≥ 1; and GRMa where a > -1. Our lower bound holds for Ma;b where a ≥ 0 and b ≥ -a; SOa;b where a; b ≥ 0 or a; b = 0; AZIa;b where a ≥ -2 and b ≥ 0; Xa and Ra where a ≥ 0; and GRMa where a > -2. [ABSTRACT FROM AUTHOR]

The connectivity index (CI), Wiener index (WI), and Randic index (RI) play a significant role in modeling various problems such as networking, transportation, and precise location detection. In this research, we explore these three indices within the framework of picture fuzzy graphs (PFGs). Initially, we present the definitions of connectivity index (CI) and Average Connectivity index (ACI) for picture fuzzy graphs. The connectivity index (CI) of a picture fuzzy graph proves to be particularly effective in addressing real-world problems related to precise location detection. As an illustration, in Sect. 4.1, we utilize the connectivity index (CI) of a picture fuzzy graph to identify the most suitable location for a school. Subsequently, we introduce the concepts of Wiener index (WI), Average Wiener index (AWI), and Hyper-Wiener index (WWI) for picture fuzzy graphs (PFGs). In Sect. 4.2, we demonstrate the application of the Wiener index in modeling intra-city journeys. Additionally, we discuss the important characteristics of these topological indices for picture fuzzy graphs. Furthermore, we investigate the relationship between the Randic index (RI) of a picture fuzzy graph and the Randic index of its subgraph, noting that the former is always greater than or equal to the latter. [ABSTRACT FROM AUTHOR]

Let G = (V (G) , E (G)) denote a graph, many important topological indices can be defined as T I (G) = ∑ v u ∈ E (G) f (d v , d u). In this paper, we study these kinds of topological indices in random spiro chains via a martingale approach. In which their explicit analytical expressions of the exact distribution, expected value, and variance are obtained. As n goes to ∞ , the asymptotic normality of topological indices of a random spiro chain is established through the Martingale Central Limit Theorem. In particular, we compute the Nirmala, Sombor, Randić, and Zagreb index for a random spiro chain along with their comparative analysis. [ABSTRACT FROM AUTHOR]

Graph theory has applications in various fields due to offering important tools such as topological indices. Among the topological indices, the Randić index is simple and of great importance. The Randić index of a graph 𝒢 can be expressed as R(G)=∑xy∈Y(G)1τ(x)τ(y)R\left( G \right) = \sum\nolimits_{xy \in Y\left( G \right)} {{1 \over {\sqrt {\tau \left( x \right)\tau \left( y \right)} }}}, where 𝒴(𝒢) represents the edge set and τ(x) is the degree of vertex x. In this paper, considering the importance of the Randić index and applications two-trees graphs, we determine the first two minimums among the two-trees graphs.

Chemical graph theory assumes an imperative part in displaying and planning any chemical structure or chemical network. The molecular structure of a chemical molecule strongly relates to its properties. The graphs that make up the molecular structure are made up of units known as vertices, and the covalent bonds that connect them are known as edges. The topological indices of any chemical compound can be used to better understand the chemical structure and its biological characteristics. In this study, the degree‐based multiplicative topological indices of the crystalline structure of cubic carbon (CCS) for levels m = 3–10 are computed, CCS[m] being one of the precious carbon allotropes. [ABSTRACT FROM AUTHOR]

In this work, we determine the maximum general Randić index (a general symmetric function of vertex degrees) for η0≤η<0 among all n-vertex unicyclic graphs with a fixed maximum degree Δ and the maximum and the second maximum general Randić index for η0≤η<0 among all n-vertex unicyclic graphs, where η0≈−0.21. We establish sharp inequalities and identify the graphs attaining the inequalities. Thereby, extremal graphs are obtained for the general Randić index, and certain open gaps in the theory of extremal unicyclic graphs are filled (some open problems are provided). We use computational software to calculate the Randić index for the chemical trees up to order 7 and use the statistical (linear regression) analysis to discuss the various applications of the Randić index with the physical properties of drugs on the said chemical trees. We show that the Randić index is better correlated with the heat of vaporization for these alkanes.

In 1997, Sierpinski graphs, S(n,k), were obtained by Klavzar and Milutinovic. The graph S(1,k) represents the complete graph Kk and S(n,3) is known as the graph of the Tower of Hanoi. Through generalizing the notion of a Sierpinski graph, a graph named a generalized Sierpinski graph, denoted by Sie(Λ,t), already exists in the literature. For every graph, numerous polynomials are being studied, such as chromatic polynomials, matching polynomials, independence polynomials, and the M-polynomial. For every polynomial there is an underlying geometrical object which extracts everything that is hidden in a polynomial of a common framework. Now, we describe the steps by which we complete our task. In the first step, we generate an M-polynomial for a generalized Sierpinski graph Sie(Λ,t). In the second step, we extract some degree-based indices of a generalized Sierpinski graph Sie(Λ,t) using the M-polynomial generated in step 1. In step 3, we generate the entropy of a generalized Sierpinski graph Sie(Λ,t) by using the Randić index.

The Randić index of a graph G is the sum of (dG(u)dG(v))−12 over all edges uv of G, where dG(u) denotes the degree of vertex u in G. In this paper, we investigate a few graph transformations that decrease the Randić index of a graph. By applying those transformations, we determine the minimum Randić index on tricyclic graphs and characterize the corresponding extremal graphs.

In this article, we study the degree-based topological indices in a random polyomino chain. The key purpose of this manuscript is to obtain the asymptotic distribution, expected value and variance for the degree-based topological indices in a random polyomino chain by using a martingale approach. Consequently, we compute the degree-based topological indices in a polyomino chain, hence some known results from the existing literature about polyomino chains are obtained as corollaries. Also, in order to apply the results, we obtain the expected value of several degree-based topological indices such as Sombor, Forgotten, Zagreb, atom-bond-connectivity, Randić and geometric-arithmetic index of a random polyomino chain.

The Randić index R(G) of a graph G is the sum of the weights (dudv)1/2 of all edges uv in G, where du denotes the degree of vertex u. Du and Zhou [On Randić indices of trees, unicyclic graphs, and bicyclic graphs, Int. J. Quantum Chem. 111 (2011), 2760-2770] determined the n-vertex unicyclic graphs with the third maximum for n ≥ 5, the fourth maximum for n ≥ 7 and the fifth maximum for n ≥ 8. Recently, Li et al. [The Randić indices of trees, unicyclic graphs and bicyclic graphs, Ars Comb. 127 (2016), 409-419] obtained the n-vertex unicyclic graphs with the sixth maximum and the seventh maximum for n ≥ 9 and the eighth maximum for n ≥ 10. In this paper, we characterize the n-vertex unicyclic graphs with the ninth maximum, the tenth maximum, the eleventh maximum, the twelfth maximum and the thirteenth maximum of Randić values. [ABSTRACT FROM AUTHOR]

Several topological indices and their chemical applicability have been studied in chemical graph theory. Some of the degree based topological indices, namely Zagreb index, Modified Zagreb index, Randic index, Atom-bond connectivity index, the fourth version of atom-bond connectivity index, Geometric arithmetic index, the fifth version of geometric arithmetic index, Sum connectivity index have been obtained for Cayley tree Γkn for k = 2. In this paper, we have computed these topological indices for the line graph of a Cayley tree L(Γkn). [ABSTRACT FROM AUTHOR]

In chemical graph theory, graph entropy is an essential parameter. It measures chemical graphs and complicated networks. In discrete mathematics, biology, and chemistry, graph entropy measures have specialized chemical uses. The purpose of this paper is to discuss graph entropies and the structure of Titanium dioxide (TiO2). We developed a new idea of computing entropies of titanium dioxide based on the degree of end vertices. Moreover, we have computed graph entropy indices for real systems, and made correlations with thermodynamic quantities like heat of formation and (thermodynamic) entropy. Also, curves have been fitted between entropy indices and thermodynamics properties (entropy, enthalpy) rational method using MATLAB. [ABSTRACT FROM AUTHOR]

Let G be a simple graph with vertex set V(G) and edge set E(G). The atom-bond connectivity and Randić index of graph G are defined as and , respectively, where du denotes the degree of vertex u in G. Let ABC – R denotes the difference between ABC index and R index. In this paper, we present a further ordering for the ABC – R indices, and determine the fourth, the fifth, the sixth, the seventh and the eighth maximum binary trees with respect to the ABC – R values. [ABSTRACT FROM AUTHOR]

Tuberculosis (TB) is one of the deadliest infectious diseases which is caused by the Mycobacterium tuberculosis. A group of first line anti-TB drugs namely isoniazid, rifampicin, pyrazinamide and ethambutol have been accepted for the treatment of drug susceptible tuberculosis. A topological index is a molecular structure descriptor which is a numerical value associated with chemical constitution for correlation of a chemical structure with its properties. The aim of this work is to investigate some degree-based topological indices for the above-mentioned anti-TB drugs using a polynomial approach. The molecular graphs of these anti-TB drugs structures were used to derive the M-Polynomials and then the derived formulas were used to calculate topological indices of the respective structures. This research could facilitate the design of new medicines against these pathogenic bacteria. [ABSTRACT FROM AUTHOR]

The activity of a molecule is related to its structure, and thus any mathematical or topological technique that can specify the underlying topological building of a molecule is important and useful. Recently, mathematical techniques have been used to characterize their structures, topologically, to find statistical correlations between their structures and drug activity. Among them, topological indices and their developments have absorbed significant consideration from both mathematical and medicinal chemists. Sugammadex is a relaxant binding agent that can be sought after the management of the aminosterioid non-depolarizing neuromuscular-blocking factors. The role of topological indices in drug development research is updated. A series of definitions in the fields of topological indices and drug discovery technologies are introduced. The aim of this paper is to investigate several well-known topological descriptors for Sugammadex. [ABSTRACT FROM AUTHOR]

In this paper, we study Young Fibonacci graphs Gn, a special family of graphs that are constructed with the help of integer partitions. Young diagrams are also used in the construction of graphs. The family of graphs is rich in structure. Thus, we investigate various properties of the family of graphs which include degree based structure and topological in-dices. Topological indices like Zagreb Index, Wiener Index, Randic Index and Connective Eccentricity Index of these graphs are computed. We also study the eigenvalues and energy of the graph. [ABSTRACT FROM AUTHOR]

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.

We assume that D is a directed graph with vertex set V(D)={v1,…vn} and arc set E(D). A VDB topological index φ of D is defined asφ(D)=12∑uv∈E(D)φdu+,dv−, where du+ and dv− denote the outdegree and indegree of vertices u and v, respectively, and φi,j is a bivariate symmetric function defined on nonnegative real numbers. Let Aφ=Aφ(D) be the n×n general adjacency matrix defined as [Aφ]ij=φdvi+,dvj− if vivj∈E(D), and 0 otherwise. The energy of D with respect to a VDB index φ is defined as Eφ(D)=∑i=1nσi(Aφ), where σ1(Aφ)≥σ2(Aφ)≥⋯≥σn(Aφ)≥0 are the singular values of the matrix Aφ.We will show that in case φ=R is the Randić index, the spectral norm of AR is equal to 1, and rank of AR is equal to rank of the adjacency matrix of D. Immediately after, we illustrate by means of examples, that these properties do not hold for most well-known VDB topological indices. Taking advantage of nice properties the Randić matrix has, we derive new upper and lower bounds for the Randić energy ER in digraphs. Some of these generalize known results for the Randić energy of graphs. Also, we deduce a new upper bound for the Randić energy of graphs in terms of rank, concretely, we show that ER(G)≤rank(G) for all graphs G, and equality holds if and only if G is a disjoint union of complete bipartite graphs.

The Randić index is among the most famous degree-based topological indices in chemical graph theory. It was introduced due to its application in modeling the properties of certain molecular structures and has been extensively studied. In this paper, we study the lower bound of the Randić index of trees in terms of the order and the total domination number. Finally, trees with the minimal Randić index are characterized. [ABSTRACT FROM AUTHOR]

Chemical descriptors are numeric numbers that capture the whole graph structure and comprise a basic chemical structure. As a topological descriptor, it correlates with certain physical aspects in addition to its chemical representation of underlying chemical substances. In the modelling and design of any chemical network, the graph is important. A number of chemical indices have been developed in theoretical chemistry, including the Wiener index, the Randić index, and many others. In this paper, we look at the benzenoid networks and calculate the exact topological indices based on the degrees of the end vertices. [ABSTRACT FROM AUTHOR]

The biological, chemical, and structural properties of molecular graphs can be predicted by using topological indices. The lower and upper bounds of fundamental topological indices are well-studied. In this paper, by using comparison theorem for integrals, we obtain new bounds for some topological indices namely, Randić index, Sombor index, harmonic index and forgotten index. [ABSTRACT FROM AUTHOR]

Connectivity can be used to measure the strength and combined power of a connected network system. Randic index of graph is one such parameter and it can measure the total combined power of a connected graphical transmission system. For two opposite sided opinion of vertices as well as edges in a bipolar fuzzy graph, it can measure the uncertainty of vertices and edges along positive and negative sides. In this article, the Randic index of bipolar fuzzy graph and bipolar fuzzy subgraph are introduced with their properties. The upper and lower boundaries of Randic index of bipolar fuzzy graphs are studied with some isomorphic properties. Randic index of directed bipolar fuzzy graphs are introduced. Several formula's are presented to calculate the Randic index of different types of regular bipolar fuzzy graphs and bipolar fuzzy cycles. Finally, two real life applications of Randic index in bipolar fuzzy graphs are described. [ABSTRACT FROM AUTHOR]

DP-coloring as a generalization of list coloring was introduced recently by Dvořák and Postle. The D P -chromatic number of G , denoted by χ D P (G) , is the minimum number k such that G is D P − k -colorable. The variation of the Randić index R ′ (G) of a graph G is defined as the sum of the weights 1 m a x { d (u) , d (v) } of all edges u v of G , where d (u) is the degree of the vertex u in G. In this paper, we show that for any graph G of order n , χ D P (G) ≤ 2 R ′ (G) , and this bound is sharp for all n and 2 ≤ χ D P (G) ≤ n. [ABSTRACT FROM AUTHOR]

A nonlinear structure–property model is proposed to predict the octane ratings of alkyl-substituted isomeric hydrocarbons of a series of cycloalkanes, which are components of gasolines that are raw materials and products of isomerization, catalytic reforming, and catalytic cracking. The model relates the octane ratings of cycloaclans to the topological indices of molecular graphs: the Wiener and Randić indices, and the sum of squares of eigen values ​​of adjacency matrix. Regression multivariate models adequately describe the octane rating of cycloalkanes. The average relative error for the octane rating for a series of 22 compounds was 2.42 units for the research method and 2.12 units for the motor method. The proposed model can be used to predict the octane ratings of cycloalkanes in gasoline components. [ABSTRACT FROM AUTHOR]

The Randić index of a graph G, denoted by R (G) , is defined as the sum of 1 / d (u) d (v) for all edges u v of G, where d (u) denotes the degree of a vertex u in G. In this note, we show that R (L (T)) > n 4 for any tree T of order n ≥ 3 . A number of relevant conjectures are proposed. [ABSTRACT FROM AUTHOR]

The graph energy E(G) of a simple graph G is sum of its absolute eigenvalues where eigenvalues of adjacency matrix A(G) are referred as eigenvalues of graph G. Depends upon eigenvalues of different graph matrices, several graph energies has been observed recently such as maximum degree energy, Randic´ energy, sum-connectivity energy etc. Depending on the definition of a graph matrix, the graph energy can be easily determined. This article contains upper bounds of several graph energies of s-regular subdivision graph S(G). Also various graph energies of complete graph are mentioned in this article.