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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. [ABSTRACT FROM AUTHOR]

Entropy is a thermodynamic function used in chemistry to determine the disorder and irregularities of molecules in a specific system or process. It does this by calculating the possible configurations for each molecule. It is applicable to numerous issues in biology, inorganic and organic chemistry, and other relevant fields. Metal–organic frameworks (MOFs) are a family of molecules that have piqued the curiosity of scientists in recent years. They are extensively researched due to their prospective applications and the increasing amount of information about them. Scientists are constantly discovering novel MOFs, which results in an increasing number of representations every year. Furthermore, new applications for MOFs continue to arise, illustrating the materials' adaptability. This article investigates the characterisation of the metal–organic framework of iron(III) tetra-p-tolyl porphyrin (FeTPyP) and CoBHT (CO) lattice. By constructing these structures with degree-based indices such as the K-Banhatti, redefined Zagreb, and the atom-bond sum connectivity indices, we also employ the information function to compute entropies. [ABSTRACT FROM AUTHOR]

The l-generalized quasi tree is a graph G for which we can find W ⊂ V (G) with | W | = l such that G − W is a tree but for an arbitrary Y ⊂ V (G) with | Y | < l , G − Y is not a tree. In this paper, inequalities with respect to zeroth-order Randić and hyper-Zagreb indices are studied in the class of l-generalized quasi trees. The corresponding extremal graphs corresponding to these indices in the class of l-generalized quasi trees are also obtained. In addition, we carry QSPR analysis of COVID-19 drugs with zeroth-order Randić and hyper-Zagreb indices (energy). [ABSTRACT FROM AUTHOR]

There are many different types of topological indices that have been known to exist with vast literature based on them. First of its kind (topological index) was introduced in 1947 by Harry Wiener and it also occurred in an approximate formula for the total π -electron energy of the molecule and the second Zagreb index appeared within the study of molecular branching. Recently, the first reverse Zagreb alpha, beta and second reverse Zagreb indices of a simple connected graph G are defined. The main object of this work is to study the reverse first and second Zagreb indices of the F-sum of two graphs G and H, denoted by G + FH in terms of the other indices of the original graphs G and H. [ABSTRACT FROM AUTHOR]

Sierpinski networks constitute an extensively studied class of graphs of fractal nature applicable in topology, mathematics of Tower of Hanoi, computer science, and Chemical graph theory. Recently the properties of Sierpinski networks have been studied like physico-chemical properties, thermodynamic properties, chemical activity, biological activity, metric dimension and degree based topological indices. In this paper, we further extend the study of Sierpinski networks and computed Zagreb types molecular descriptors (molecular topological indices) and their variants. [ABSTRACT FROM AUTHOR]

Topological indices are the atomic descriptors that portray the structures of chemical compounds and they help us to anticipate certain physico-compound properties like boiling point, enthalpy of vaporization and steadiness. These properties can be described by certain graph invariants alluded to as topological indices. In this paper, we have computed topological indices of Para-line graph for honeycomb and graphene systems. [ABSTRACT FROM AUTHOR]

The relations between the physico‐chemical properties of a chemical compounds its molecular structure properties are used in quantitative structure activity and property relationship studies by using graph‐theoretical techniques. The Wiener polarity index is the number of unordered pairs of vertices lying at distance 3 in a graph. This index is correlated to the cluster coefficient of chemical networks. The Wiener polarity index has been used to exhibit quantitative structure–property relationships in a series of acyclic and cycle‐containing hydrocarbons. In this paper, we consider three variants of the graph of titanium oxide TiO2, that is, two‐dimensional lattice, nanotubes and nanotorus. For all these graphs, we compute the number of pairs of vertices lying at distance one, two and three. Using this information, we compute the Wiener polarity index and leap Zagreb indices of these graphs. [ABSTRACT FROM AUTHOR]

There are various methods available which are used to search large chemical databases and to predict the physicochemical properties of molecular structures. Using molecular descriptors for this purpose is the simplest of these methods. The Zagreb indices are amongst the oldest molecular descriptors, and their properties have been extensively studied and applied in QSAR/QSPR studies. The Zagreb coindices were recently introduced, attracting the attention of researchers in mathematical chemistry. In this paper, we study Zagreb indices and several other Zagreb-type indices including the general Randić index, sum-connectivity index, F-index, and Zagreb coindices of R -vertex and edge join of two arbitrary graphs. [ABSTRACT FROM AUTHOR]

In a connected graph G with a vertex v , the eccentricity ε v of v is the distance between v and a vertex farthest from v in the graph G. Among eccentricity-based topological indices, the eccentric connectivity index, the total eccentricity index, and the Zagreb index are of vital importance. The eccentric connectivity index of G is defined by ξ G = ∑ v ∈ V G d v ε v , where d v is the degree of the vertex v and ε v is the eccentricity of v in G. The topological structure of an interconnected network can be modeled by using graph explanation as a tool. This fact has been universally accepted and used by computer scientists and engineers. More than that, practically, it has been shown that graph theory is a very powerful tool for designing and analyzing the topological structure of interconnection networks. The topological properties of the interconnection network have been computed by Hayat and Imran (2014), Haynes et al. (2002), and Imran et al. (2015). In this paper, we compute the close results for eccentricity-based topological indices such as the eccentric connectivity index, the total eccentricity index, and the first, second, and third Zagreb eccentricity index of a hypertree, sibling tree, and X -tree for k -level by using the edge partition method. [ABSTRACT FROM AUTHOR]

A variety of dendrimers exist, and each has biological properties such as polyvalency, selfassembling, electrostatic interactions, chemical stability, low cytotoxicity, and solubility. These varied characteristics make dendrimers a good choice in the medical field and pharmaceuticals. A topological index is actually designed by transforming a molecular structure into a numeric number. These topological indices correlate certain physicochemical properties like boiling point, stability, strain energy etc of chemical compounds. It is reported that these indices are useful in the study of anti-in ammatory activities, anticancer therapies and diagnostic imaging of certain chemical networks. In this paper, we determine hyper Zagreb index, first multiple Zagreb index, Second multiple Zagreb index, Zagreb polynomials for nanostar dendrimers, Benzenoid and phenylenes. [ABSTRACT FROM AUTHOR]

Chemical graph theory helps to understand the structural properties of a molecular graph. The molecular graphs are the graphs that consist of atoms called vertices and the covalent bond between them called edges. The eccentricity u of vertex u in a connected graph G, is the distance between u and a vertex farthermost from u. In this article, we have introduced degree eccentric-connectivity index and degreev eccentric-connectivity polynomial and computed these for the crystal structure of cubic carbon. Moreover we have also computed eccentric Zagreb indices, eccentric based atom bond connectivity index and eccentric based geometric arithmetic index of crystal structure of cubic carbon for CCC(n). [ABSTRACT FROM AUTHOR]

Graph theory has made much advancement in the field of mathematical chemistry in last few decades. Now a days, chemical graph theory has become very popular among researchers because of its wide range of applications in mathematical chemistry. The molecular topological descriptors are the numerical invariants of a molecular graph and are very useful for predicting their bioactivity. A great variety of such indices are studied and used in theoretical chemistry, pharmaceutical researchers, in drugs and in different other fields. In this article, we study the chemical graph of honeycomb network denoted by HCn and compute the total eccentricity index, average eccentricity index, eccentricity based Zagreb indices, atom-bond connectivity index and geometricarithmetic index of honeycomb networks. Furthermore, we give analytically closed formulas of these indices which are helpful in studying the underlying topologies of this netwrok. [ABSTRACT FROM AUTHOR]

In the studies of quantitative structure–activity relationships (QSARs) and quantitative structure–property relationships (QSPRs), graph invariants are used to estimate the biological activities and properties of chemical compounds. In these studies, degree-based topological indices have a significant place among the other descriptors because of the ease of generation and the speed with which these computations can be accomplished. In this paper, we give the results related to the first, second, and third Zagreb indices, forgotten index, hyper Zagreb index, reduced first and second Zagreb indices, multiplicative Zagreb indices, redefined version of Zagreb indices, first reformulated Zagreb index, harmonic index, atom-bond connectivity index, geometric-arithmetic index, and reduced reciprocal Randić index of a new graph operation named as "subdivision vertex-edge join" of three graphs. [ABSTRACT FROM AUTHOR]

Graph theory is vital for modeling and designing any chemical structure or network. Chemical graph theory is helpful to under first and about the structural characteristics of a molecular graph. The molecular graph is a graph in which vertices represent atoms and edges represent the chemical bond between atoms. Among topological descriptors, connectivity invariants are very beneficial and they have a useful part in chemistry. In this article, we study the chemical graphs of carbon graphite and we compute and derive close formulas of eccentricity based topological indices, such as eccentric based first Zagreb index, eccentric based second Zagreb index and eccentric based third Zagreb index of carbon graphite CG(m; n) for 3-layer. [ABSTRACT FROM AUTHOR]