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In mathematical chemistry, the algebraic polynomial serves as essential for calculating the most accurate expressions of distance-based, degree-distance-based, and degree-based topological indices. The chemical reactivity of molecules, which includes their tendency to engage in particular chemical processes or go through particular reactions, can be predicted using topological indices. Considerable effort has been put into examining the many topological descriptors of simple graphs using ring structures and well-known groups instead of nonassociative algebras, quasigroups, and loops. Both finite quasigroups and loops are the generalizations of groups. In this article, we calculate topological descriptors and some well-known polynomials, M -polynomial, Hosoya's polynomial, Schultz's polynomial, and modified Schultz polynomial of finite relatively prime graphs of most orders connected with two classes of quasigroups and go through their graphical aspects. [ABSTRACT FROM AUTHOR]

Metal–organic frameworks (MOFs) are vital in modern material science, offering unique properties for gas storage, catalysis, and drug delivery due to their highly porous and customizable structures. Chemical graph theory emerges as a critical tool, providing a mathematical model to represent the molecular structure of these frameworks. Topological indices/molecular descriptors are mathematical formulations applied to molecular models, enabling the analysis of physicochemical properties and circumventing costly lab experiments. These descriptors are crucial for quantitative structure-property and structure-activity relationship studies in mathematical chemistry. In this paper, we study the different molecular descriptors of tetracyanobenzene metal–organic framework. We also give numerical comparison of computed molecular descriptors. [ABSTRACT FROM AUTHOR]

A topological index is a numerical parameter that represents the molecular structure of a compound based on its graph‐theoretical properties. It provides a means to describe the structure‐activity relationship, as well as the physicochemical properties of the compound. Topological indices are used in various scientific fields, including chemistry, biology, and computer science, to study and predict the behavior of molecules. In this study, we concentrate on the calculation of M‐polynomials and entropy for the allotropic form of carbon, diamond. Algebraic operations are applied to these polynomials to obtain degree‐dependent topological invariants. [ABSTRACT FROM AUTHOR]

The molecular graph theory is one of the essential tools for developing and constructing molecular structure. In particular, using topological invariant, which is a numerical descriptor of molecular graph, has become widespread in predicting physical properties. Topological indices provide knowledge about the overall form of molecular graphs, particularly various degree-based topological indices. This research paper focuses on the first and second Zagreb index and their coindices for the line graph of a transition metal-organic network. The network, denoted by L (TM − TCNB) , comprises the transition metals Ti, V, Cr, Fe, Co, Ni, Cu or Zn, and the TCNB ligand. The line graph of a network is a graph that represents the connections between the edges of the original network. The Zagreb indices are widely used to describe the molecular graph's shape and structure. The paper highlights the importance of these indices in modelling the chemical properties of the transition metal-organic network. The study results show that the Zagreb indices and their coindices strongly correlate with several physical characteristics, for instance the boiling points, melting points, and density of the network. [ABSTRACT FROM AUTHOR]

L-valine is considered one of the essential amino acids for mammals, which belongs to the glycogen types of amino acids. This amino acid is valuable for making protein. Due to the diverse structure of N-salicylidene-L-valine and its higher catalytic and cytotoxic activity it gains impressive interest. To study this structure more deeply and to know more about its properties in terms of topological indices, we study the supramolecular chain in dialkyltin of complex- 2 , 3 , and 4 of N-salicylidene-L-valine. In the end, we provide a comparative study of these structures by taking particular examples and also gives a comparison between topological indices. [ABSTRACT FROM AUTHOR]

Due to the recent economic crisis of the world, producing low-cost objects in every field of science and engineering is the main topic of research. Making small in-sized objects, low in budget and weight, and efficient in quality is necessary for the present and future world. In this whole scenario, nanotechnology contributing more than enough. Nanomaterials and nano-objects are highly preferable, carbon nanotubes are one of the highly chosen structures of nanomaterials, which is used in different chemical, biological, and technological fields. This thought leads us to study its different aspects of physical and chemical properties. Topological descriptors are used to characterize different molecular structures by making its numerical quantitative analysis. This article deals with different carbon nanotube structures, studying their chemical and physical characterizations and comparing armchair, semicapped, and capped carbon nanotubes. It provides the best option by making comparative analysis with different tools and algorithms. [ABSTRACT FROM AUTHOR]

In this paper, we discuss the degree based topological properties of the novel planar metal-organic networks TM – TCNB. Interestingly, the TM – TCNB systems are metallic in any event in one turn heading and show long-run ferromagnetic coupling on the off chance that for attractive structures, which speak to perfect applicants and a fascinating possibility of uncommon applications in spintronics. Chemical graph theory plays an important role in modeling and designing any chemical structure. The topological indices are the numerical invariants of a molecular graph and are very useful for predicting their physical properties. We have computed the degree based topological indices for this metal-organic networks TM – TCNB. [ABSTRACT FROM AUTHOR]

G. Polya laid the foundation of counting polynomials in chemical graph theory. These polynomials can be associated with various graphs to signify their properties. In this study, a new concept has been introduced to construct the counting polynomials such as, Sadhana Theta and PI polynomials in terms of omega polynomial. A star expression has been established in order to transform equidistant into non equidistant edges. Furthermore, new relations have also been evolved to calculate Theta, PI and Sadhana Indices through Omega polynomial. [ABSTRACT FROM AUTHOR]

Recent findings of diverse forms of carbon nanostructures have encouraged research on their applications in several areas. They are used as a structural material in electronic and optical devices, involving transistors, photodetectors, plastic, biological sensors, pharmaceutical and medicine, and drug delivery systems. Since the discovery of Y‐shaped carbon nanotubes (CNTs) junctions, they have drawn interest for future electron devices, for example, three‐terminal transistors, amplifiers, and switches. A topological index is a number that is used to study the physicochemical properties of chemical structures. In this paper, we are providing topological properties of CNT Y‐junctions. The results obtained for different types of Y‐junction graphs appeared with comparisons. According to the best of our knowledge and literature, this is the first report about topological indices on this class of CNT junctions. [ABSTRACT FROM AUTHOR]

A graph in which atoms are taken as vertices and bonds can be shown by edges is called a molecular graph. For such graphs we can find the indices showing their bioactivity as well as their physio-chemical properties. Industry is flourishing nowadays just because of the interdisciplinary study of different subjects. This gives us a way to understand the application of different subjects. Chemical graph theory is a mixture of chemistry and mathematics both play an important role in chemical graph theory. Chemistry provides a chemical compound and graph theory transform this chemical compound into a molecular graph which further studied by different aspects such as topological indices. In this article, we computed generalized Randić, general Zagreb, general sum-connectivity, ABC, GA, A B C 4 , and G A 5 indices of line graphs of the H A C 5 C 7 [ p , q ] and V C 5 C 7 [ p , q ] nanotubes." [ABSTRACT FROM AUTHOR]

In this paper, we computed the edge version of atom-bond connectivity and geometric arithmetic indices of H-Pantacenic nanotube. We also computed ABC4 and GA5 indices of the line graph of H-Pantacenic nanotube. [ABSTRACT FROM AUTHOR]

In the study of QSAR/QSPR, topological indices such as Shultz index, generalized Randic index, Zagreb index, general sum-connectivity index, atom-bond connectivity (ABC) index and geometric–arithmetic (GA) index are exploited to estimate the bioactivity of chemical compounds. A topological index attaches a chemical structure with a numeric number. There are numerous applications of graph theory in this field of research. In this paper we computed generalized Randic, general Zagreb, general sum-connectivity, ABC, GA, ABC 4 and GA 5 indices of the line graphs of 2 D -lattice, nanotube and nanotorus of TUC 4 C 8 [ p, q ] by using the concept of subdivision. [ABSTRACT FROM AUTHOR]

In QSAR/QSPR study, topological indices such as Shultz index, generalized Randic index, Zagreb index, general sum-connectivity index, atom-bond connectivity (ABC) index and geometric-arithmetic (GA) index are utilized to guess the bioactivity of chemical compounds. A topological index in fact relates a chemical structure with a numeric number. Graph theory has established a significant use in this area of research. In this paper we computed ABC 4 and GA 5 indices of the line graph of tadpole, wheel and ladder graphs using the notion of subdivision. [ABSTRACT FROM AUTHOR]

Topological indices are empirical features of graphs that characterize the topology of the graph and, for the most part, are graph independent. An important branch of graph theory is chemical graph theory. In chemical graph theory, the atoms corresponds vertices and edges corresponds covalent bonds. A topological index is a numeric number that represents the topology of underline structure. In this article, we examined the topological properties of prism octahedron network of dimension m and computed the total eccentricity, average eccentricity, Zagreb eccentricity, geometric arithmetic eccentricity, and atom bond connectivity eccentricity indices, which are used to utilize the distance between the vertices of a prism octahedron network. [ABSTRACT FROM AUTHOR]

There are different mathematical tools to establish a numeric correlation between a chemical structure and its properties. A topological index is such a successful tool. It represents a graph in form of a single real number. In QSAR/QSPR study, the bioactivity of chemical compounds is predicted, using physico-chemical properties and topological indices. In this area of research, graph theory plays a vital role. The four operations on graphs were defined by Eliasi and Taeri in 2009. Imran et al. studied Cartesian product of the four operations on graph and computed bounds for some degree-based topological indices in 2017. In this paper, we extend this study for the lexicographic product of graphs to determine bounds of Zagreb indices, multiple Zagreb indices, and F-index for four sums of connected graphs. [ABSTRACT FROM AUTHOR]

In this work, we study the degree-based topological invariants, and the general sum-connectivity, A B C 4 , G A 5 , general Zagreb, G A , generalized Randić, and A B C indices of the line graphs of some rooted product graphs ( C n { P k } and C n { S m + 1 } ) are determined by menas of the concept of subdivision. Moreover, we also computed all these indices of the line graphs of the subdivision graphs of i-th vertex rooted product graph C i , r { P k + 1 } . [ABSTRACT FROM AUTHOR]

A branch of graph theory that makes use of a molecular graph is called chemical graph theory. Chemical graph theory is used to depict a chemical molecule. A graph is connected if there is an edge between every pair of vertices. A topological index is a numerical value related to the chemical structure that claims to show a relationship between chemical structure and various physicochemical attributes, chemical reactivity, or, you could say, biological activity. In this article, we examined the topological properties of a planar octahedron network of m dimensions and computed the total eccentricity, average eccentricity, Zagreb eccentricity, geometric arithmetic eccentricity, and atom bond connectivity eccentricity indices, which are used to determine the distance between the vertices of a planar octahedron network. [ABSTRACT FROM AUTHOR]