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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]

Phenylenes belong to a special class of conjugated hydrocarbons composed of a special arrangement of hexagons and squares. In the phenylene structure, every square is adjacent to a pair of hexagons, and any two hexagons are not adjacent. If in the structure of phenylene only two squares are adjacent to every hexagon, then it is called a phenylene chain. In this paper, we compute the expected values of the first and second Gourava index, the first and second redefined Zagreb index, and the Hyper-Zagreb index of random phenylene chains. Furthermore, a comparison between the expected values of the computed indices among the classes of random phenylene chains has been investigated. [ABSTRACT FROM AUTHOR]

Boron clusters are small, three-dimensional structures made of boron atoms. These come in a wide range of shapes and sizes, from basic planar constructions to complex polyhedral patterns. Boron clusters are fascinating from both a fundamental and practical perspective because of their unique electrical, optical, and chemical properties. More focus has lately been placed on boron cluster sheets, which are two-dimensional structures made of boron clusters. Quantitative structure-activity relationship (QSAR) studies implement M-polynomials, a type of molecular descriptor, to describe the topology of molecules. These are based on the notion of graph theory, which provides a theoretical framework for quantitatively studying molecular graphs. In this article, we discussed some novel topological descriptors-based M-polynomials and found algebraic formulations for the boron cluster or borophene sheets. We discussed first Zagreb, second Zagreb and Randić M-polynomials, based on the different differential and integral operators. [ABSTRACT FROM AUTHOR]

Totally made of carbon atoms, fullerene is a spherical molecule with a three-dimensional pattern like a soccer ball. The most prevalent kind of fullerene is C60, which has 60 carbon atoms organised in a pattern like that of a soccer ball (12 pentagons and 20 hexagons). The carbon atoms are connected by a web of covalent connections to form what is known as a truncated icosahedron. The fullerene molecule's spherical form makes it beneficial in many applications, including medication delivery, lubricants, and even as a possible component of nanotechnology. Topological indices or numerical descriptors have been used for decades to connect essential physicochemical parameters with significant molecular structural features including cyclicity, boiling and melting point, and enthalpy for a range of two- along with three-dimensional in form chemical arrangement networking or graphs. The valency of atom-based numerical descriptors in this study are computed using the valency of atom in a chemical network or structure of the molecular investigation. In chemoinformatics, the numerical descriptors based on the degree of defined by reverse-degree concept technique is a more newly developed methodology for evaluating chemical systems and geometries. It draws attention to molecular traits as numerical descriptors based on the degree of defined by reverse-degree concept and presents numerical descriptors in algebraic form. In this work, we describe several fullerene cages as numerical descriptors based on the degree of defined by reverse-degree concept and compute a number of numerical descriptors based on the degree of defined by reverse-degree concept, such as the Randić index, atom-bond-connectivity, geometric arithmetic, forgotten, hyper, first and second Zagreb indices are computed. [ABSTRACT FROM AUTHOR]