The production of low-cost, small in size, and high in efficiency objects is the topic of research in almost all scientific fields, especially of engineering. In this scenario, nanotechnology becomes of great importance. To achieve these tasks, one needs to study the different physical and chemical aspects of a chemical compound. In mathematical graph theory, chemical compounds are transformed in a unique mathematical representation and then examined under various parameters for these purposes. The Partition Dimension is also one of these parametric tools, which are used to identify each atom or vertex of a chemical structure under some chosen conditions. In the present paper, we use this tool to show the unique identification of each atom of a tri-hexagonal lattice of the $\alpha $ -boron nanotube.

Intersecting longest paths and longest cycles: A survey

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, $AB{C}_4,$ and $G{A}_5$ indices of line graphs of the $HA{C}_5{C}_7\left[p,q\right]$ and $V{C}_5{C}_7\left[p,q\right]$ nanotubes.”

In chemistry, customary depiction for the structure of chemical compounds are graphs where the vertices and edges represents the atoms and bonds types, respectively. Resolving set $\aleph$ is an ordered subset of vertices of a graph G, in which each vertex of G is distinctively determined by its distance to the vertices in $\aleph .$ The cardinality of a minimum resolving set is called the metric dimension of G. A resolving set ${\aleph }_f$ of G is fault-tolerant if ${\aleph }_f\setminus v$ is also a resolving set, for every v in ${\aleph }_f.$ Resolving set allows to obtain a unique representation for chemical structures. In particular, they were used in pharmaceutical research for discovering patterns common to a variety of drugs. In this paper we determine the exact metric and fault tolerant metric dimension of cyclic hexagonal square chain, linear phenylene, and linear heptagons structure.