Topological indices serve as mathematical tools for characterising the molecular structure of a compound, and are useful to anticipate its properties. Actually, these are quantitative measures that can provide valuable information regarding the structure of a molecule, such as its connectivity and symmetry. By analysing these indices, researchers can make predictions about the behaviour of the molecule, such as its reactivity, solubility, and toxicity, among others. The γ-Graphyne is a fascinating carbon allotrope that has recently gained significant attention due to its unique electronic, optical, and mechanical properties. As a result, there has been increasing interest in exploring its potential applications in various fields of science and technology. The molecular descriptors for the characterisation of γ-Graphyne have not yet been investigated. Therefore, it is of much importance to predict its molecular topology to well understand the physicochemical properties. In this work, a graph theory-based edge partitioning technique is used to model the molecular topology of γ-Graphyne and Zigzag graphyne nanoribbon, and mathematical closed-form expressions for some of its essential degree-based molecular descriptors are derived. These computed indices have been illustrated with the help of graphical representations and numerical tables. GRAPHICAL ABSTRACT

Mathematical chemistry is concerned with the use of mathematics to solve the problems in chemistry. A triangular chain graph is a simple graph composed of a series of triangles. This chain is widely used in the formation of triangular cactus, boron clusters, boron triangular nanotubes, and boron nanotubes. In this paper, the first Zagreb connection index, the first modified Zagreb connection index, and the second Zagreb connection index for triangular chain structures are calculated and derived closed formulas for them. Based on the derived formulas and obtained numerical results, the physicochemical properties of these type of structures can easily be investigated.

Metal–Organic Networks (MONs) are made by chemical molecules that contain metal ions and organic ligands. A crystalline porous solid called Metal–Organic Networks (MONs) is made up of a $$3D$$3 D metal network of ions held in place by a multidentate ligand. (MONs) can be used for gas storage, purification drug delivery, gas separation, catalysis, and sensing applications. There is enormous potential for effective integration and research of MONs in diverse applications. Molecular descriptors are arithmetic measures that reveal a chemical substance's physical and chemical characteristics in its foundational network in a natural relationship. They demonstrate an important role in theoretical and ecological chemistry, and in the field of medicine. In this research, we calculated various recently discovered molecular descriptors viz. the modified version of second zagreb index, harmonic index, reciprocal randic index, modified version of forgotten topological index, redefined first zagreb topological index, redefined second zagreb topological index and redefined third zagreb topological index for two separate metal–organic networks. The numerical and graphical comparative analysis of these considered molecular descriptors are also performed.