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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]

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]

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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]

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]

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]

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]

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 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]

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]

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]

A chemical compound can be represented as a chemical graph. A topological index of a (chemical) graph is a numeric value of a graph which characterize its topology and is usually graph invariant. The Zagreb indices, Randic index and sum-connectivity indices are useful in the study of antiinflammatory activities, boiling point, molecular complexity heterosystems of certain chemical instances, and in elsewhere. In this paper, we calculate the mentioned topological indices of some infinite classes of prophyrin dendrimers. [ABSTRACT FROM AUTHOR]

Some well defined connectivity topological indices are Randic index, atom-bond connectivity index, geometric-arithmetic index and Shigehalli & Kanabur indices, brought into light by M. Randic, Estrada et al, Vukicevic et al and V. S. Shigehalli, in their respective research articles. Topological indices preserve the symmetry of molecular structures and provide a mathematical formulation to predict their properties like boiling points, viscosity and the radius of gyrations,1 mainly their study gets a cover under the category of physical chemistry. Due to its mathematical nature, this idea has caught the attention of many chemists. It has also been reported that these indices are useful in the study of anti-inflammatory activities of certain chemical instances. In this paper, we shall calculate these topological indices of an infinite class of octagonal tilling structures OT [m, n], which is a molecular graph of a semiconductor allotrope consisting of octagons and rectangles, for all possible values of the parameters m and n. We shall also calculate Shigehalli & Kanabur indices of infinite structure of the titania TiO2 nanotubes. [ABSTRACT FROM AUTHOR]

Let G = ( V , E ) be a molecular graph, in which the vertices of V represent atoms and the edges of E the bonds between pairs of atoms. One of the earliest and most widely studied degree-based molecular descriptor, the Randić index of G , is defined as R ( G ) = ∑ u v ∈ E ( d u d v ) − 1 2 , where d u denotes the degree of u  ∈  V . Bollobás and Erdős (1998) generalized this index by replacing − 1 2 with any fixed real number. To facilitate the enumeration of these indices, motivated by earlier work of Dvořák et al. (2011) and Knor et al. (2015) introduced two other indices, that provide lower and upper bounds, respectively: R α ′ ( G ) = ∑ u v ∈ E min { d u α , d v α } and R α ″ ( G ) = ∑ u v ∈ E max { d u α , d v α } . In this paper, we give expressions for computing R α ′ and R α ″ of benzenoid systems and phenylenes, as well as a relation between R α ′ and R α ″ of a phenylene and its corresponding hexagonal squeeze. We also determine the extremal values of R α ′ and R α ″ in benzenoid systems with h hexagons for different intervals for the value of α . [ABSTRACT FROM AUTHOR]

Let G be a graph with edge set E ( G ). The Randić and sum-connectivity indices of G are defined as R ( G ) = ∑ u v ∈ E ( G ) 1 d e g G ( u ) d e g G ( v ) and S C I ( G ) = ∑ u v ∈ E ( G ) 1 d e g G ( u ) + d e g G ( v ) , respectively, where deg G ( u ) denotes the vertex degree of u in G . In this paper, the extremal Randić and sum-connectivity index among all n-vertex chemical trees, n  ≥ 13, connected chemical unicyclic graphs, n  ≥ 7, connected chemical bicyclic graphs, n  ≥ 6 and connected chemical tricyclic graphs, n  ≥ 8, were presented. [ABSTRACT FROM AUTHOR]

This article is devoted to establishing some extremal results with respect to the difference of two well-known bond incident degree indices [atom-bond connectivity ( ABC) index and Randić ( R) index] for the chemical graphs representing alkanes. More precisely, the first three extremal trees with respect to ABC - R are characterized among all n-vertex binary trees (the trees with maximum degree at most 3). The n-vertex chemical trees (the trees with maximum degree at most 4) having the first three maximum ABC - R values are also determined. [ABSTRACT FROM AUTHOR]

The Randic Connectivity Index R(G) is one of the oldest connectivity index, introduced by Randic in 1975. Another connectivity indices is the Sum-Connectivity Index X(G) introduced in 2008 by Zhou and Trinajstic. Recently in 2011, a modification of the Randic Connectivity Index of a graph G was introduced by Dvorak et al. In this paper, we compute these connectivity topological indices for a family of molecular graphs known as titania nanotubes TiO2(m,n). [ABSTRACT FROM AUTHOR]

A numerical quantity that characterizes the whole structure of a graph is called a topological index. The concept of Randić ( Rα), atom−bond connectivity (ABC), and geometric−arithmetic (GA) topological indices was established in chemical graph theory based on vertex degrees. In this paper, we study a carbon nanotube network that is motivated by the molecular structure of a regular hexagonal lattice and determine Rα, ABC, and GA indices for this important class of networks. [ABSTRACT FROM AUTHOR]

Chemical compounds are often modeled as polygonal shapes, where a vertex represents an atom and an edge symbolizes a bond. A topological index is a number related to a molecular graph invariant. In this paper, exact formulas for the eccentricic connectivity and Randić indices of hexagonal parallelogram of benzenoids are given. [ABSTRACT FROM AUTHOR]

Let G be a simple, connected and undirected graph. The atom-bond connectivity index ( A B C (G) ) and Randić index ( R (G) ) are the two most well known topological indices. Recently, Ali and Du (2017) introduced the difference between atom-bond connectivity and Randić indices, denoted as A B C − R index. In this paper, we determine the fourth, the fifth and the sixth maximum chemical trees values of A B C − R for chemical trees, and characterize the corresponding extremal graphs. We also obtain an upper bound for A B C − R index of such trees with given number of pendant vertices. The role of symmetry has great importance in different areas of graph theory especially in chemical graph theory. [ABSTRACT FROM AUTHOR]

A topological index of a graph is a single numeric quantity which relates the chemical structure with its underlying physical and chemical properties. Topological indices of a nanosheet can help us to understand the properties of the material better. This study deals with computation of degree-dependent topological indices like the Randic index, first Zagreb index, second Zagreb index, geometric arithmetic index, atom bond connectivity index, sum connectivity index and hyper Zagreb index of nanosheet covered by C3 and C6. Furthermore, M-polynomial of the nanosheet is also computed, which provides an alternate way to express the topological indices. [ABSTRACT FROM AUTHOR]