Closed Formulas for Some New Degree Based Topological Descriptors Using M-polynomial and Boron Triangular Nanotube (original) (raw)
Related papers
New degree-based topological descriptors via m-polynomial of boron α-nanotube
2020
The study of molecular structure having size less than 100 nm is called nanotechnology. Nano-materials have vast applications in different fields. Boron α-nanotube is very famous in Nano-science. In this article, we computed some important topological indices of this structure using their M-polynomial along with plotting the results.
Scientific Reports, 2023
A numeric quantity that characterizes the whole structure of a network is called a topological index. In the studies of QSAR and QSPR, the topological indices are utilized to predict the physical features related to the bioactivities and chemical reactivity in certain networks. Materials for 2D nanotubes have extraordinary chemical, mechanical, and physical capabilities. They are extremely thin nanomaterials with excellent chemical functionality and anisotropy. Since, 2D materials have the largest surface area and are the thinnest of all known materials, they are ideal for all applications that call for intense surface interactions on a small scale. In this paper, we derived closed formulae for some important neighborhood based irregular topological indices of the 2D nanotubes. Based on the obtained numerical values, a comparative analysis of these computed indices is also performed. Carbon nanotubes (CNTs) are actually cylindrical molecules that comprise of rolled-up sheets of single-layer carbon atoms (graphene). They can be single-walled having a less than 1 nm (nm) diameter or multi-walled, comprising of numerous concentrically interlinked nanotubes, with around more than 100 nm diameters. Sumio Iijima discovered the multi-walled carbon nanotubes in 1991 1. CNTs are bonded with sp 2 bonds chemically, an extremely strong form of molecular interaction. These nanotubes inherit electrical properties from graphene, which are determined by the rolling-up direction of the graphene layers. Apart from these, CNTs also have distinctive mechanical and thermal properties like lightweight , high tensile strength, low density, better thermal conductivity, high aspect ratio and high chemical stability. All these properties make them intriguing for new materials development, especially CNTs are best candidates for hydrogen storage cells, cathode ray tubes (CRTs), electronic devices, electron field emitters and transistors. Keeping in view their strong applicability and importance, it is very important to model and characterize these CNTs for a better understanding of their structural topology for enhancement of their physical properties. The study of chemicals using a mathematical method is called mathematical chemistry. Chemical graph theory is a branch of chemistry that uses graph theory concepts to convert chemical events into mathematical models. The chemical graph is a simple connected graph in which atoms and chemical bonds are taken as vertices and edges respectively. A connected graph of order n = |V (G)| and size m = |E(G)| can be created with the help of G and edge set E. The focus of research in the area of nanotechnology is on atoms and Molecules. The Cartesian product of a path graph of m and n is called a 2D lattice. Graph theory has emerged as a powerful tool for analyzing the structural properties of complex systems represented by graphs. Topological indices, which are numerical quantities derived from graph theory 2-8 , have gained significant attention due to their ability to concisely capture important graph properties. Degree-based topological indices specifically utilize the degrees of vertices in a graph to quantify its structural characteristics 9. Degree based indices, such as the Randić index, the atom-bond connectivity index, and the Harary index, capture the connectivity and branching patterns in a graph by considering the distances between pairs of vertices in relation to their degrees 10-14. These indices have found wide applications in drug design, chemical graph theory, and network analysis 15-18 .
COMPUTING SOME TOPOLOGICAL INDICES OF CARBON NANOTUBES
In this paper, several topological indices are investigated for H-Phenylenic nanotube, H-Naphthylenic nanotube and H-Anthracenic nanotube. The calculated indices are product-connectivity index, sum-connectivity index, geometric-arithmetic index and atom-bond connectivity index.
Computational Analysis of topological indices of two Boron Nanotubes
Scientific Reports
There has been a recent debate that boron nanotubes can outperform carbon nanotubes on many grounds. The most stable boron nanotubes are made of a hexagonal lattice with an extra atom added to some of the hexagons called ∝-boron nanotubes. Closed forms of M-polynomial of nanotubes produce closed forms of many degree-based topological indices which are numerical parameters of the structure and determine physico-chemical properties of the concerned nanotubes. In this article, we compute and analyze many topological indices of ∝-boron nanotubes correlating with the size of structure of these tubes through the use of M-polynomial. More importantly we make a graph-theoretic comparison of indices of two types of boron nanotubes namely triangular boron and ∝-boron nanotubes. Mathematical chemistry provides tools such as polynomials and functions to capture information hidden in the symmetry of molecular graphs and thus predict properties of compounds without using quantum mechanics. A topological index isa numerical parameter of a graph and depicts its topology. It describes the structure of molecules numerically and are used in the development of qualitative structure activity relationships (QSARs). Most commonly known invariants of such kinds are degree-based topological indices. These are actually the numerical values that correlate the structure with various physical properties, chemical reactivity and biological activities 1-5. It is an established fact that many properties such as heat of formation, boiling point, strain energy, rigidity and fracture toughness of a molecule are strongly connected to its graphical structure. Hosoya polynomial, (Wiener polynomial) 6 , plays a pivotal role in distance-based topological indices. A long list of distance-based indices can be easily evaluated from Hosoya polynomial. A similar breakthrough was obtained recently by Klavzar et. al. 7 , in the context of degree-based indices. Authors in 7 introduced M-polynomial in, 2015, to play a role, parallel to Hosoya polynomial to determine closed form of many degree-based topological indices 8-12. The real power of M-polynomial is its comprehensive nature containing healthy information about degree-based graph invariants. These invariants are calculated on the basis of symmetries present in the 2d-molecular lattices and collectively determine some properties of the material under observation. Because of increasing interests and developments of new nanomaterials, computations have minimized the burden of experimental labor to some extent. Amongst the nanomaterials, nanocrystals, nanowires and nanotubes, constitute three major categories, the last two being one-dimensional. Boron nanotubes are becoming increasingly interesting because of their remarkable properties like structural stability, work function, transport properties, and electronic structure 13. Triangular Boron is derived from a triangular sheet as shown in Fig. 1. The first boron nanotubes were created, in 2004, from a buckled triangular latticework 13-15. These tubes are discussed recently in 15,16. Other well-known type, α-boron, is derived from α-sheet. Irrespective of their structures and chiralities, both types are more conductive than carbon nanotubes 14,17-19. Figure 2 describes basic structure of ∝boron nanotube. Following Fig. 3 also presents different views of ∝-Boron nanotube, (a) is the planar view whereas (b) is the tabular view. As for as structure of both tubes are concerned, ∝-Boron nanotube is more complicated than Triangular boron nanotubes with addition of an extra atom to the center of some of the hexagons 15. In 15 , authors proved that this is the most stable known theoretical structure for a boron nanotube. They also showed that, with this pattern, boron nanotubes should have variable electrical properties: wider ones would be metallic conductors,
Due to the presence of multicenter bonds and their novel electronic properties, boron nanotubes are attractive. The tri-hexagonal boron nanotubes are build up from triangles and hexagons. It is useful to the QSPR/QSAR studies. Topological indices are classified in different forms such as, degree based topological indices, distance based topological indices and counting related topological indices etc. Here, we concentrated the reckoning of topological indices such as first zagreb, second zagreb, modified second zagreb index, generalized randic index, symmetric division degree index for the tri-hexagonal boron nanotube. Also, established their M-polynomials and using maple software plotted the 3D structure.
Computing Degree-Based Topological Indices of Polyhex Nanotubes
2016
Recently, Shigehalli and Kanabur [20] have put forward for new degree based topological indices, namely Arithmetic-Geometric index (AG1 index), SK index, SK1 index and SK2 index of a molecular graph G. In this paper, we obtain the explicit formulae of these indices for Polyhex Nanotube without the aid of a computer.
Some Formulas for the Polynomials and Topological Indices of Nanostructures
Acta Chemica Iasi, 2016
In this paper, we focus on the structure of Polycyclic Aromatic Hydrocarbons (PAHs) and calculate the Omega and its related counting polynomials of nanostructures. Also, the exact expressions for the Theta, Sadhana, Pi, Hyper Zagreb and Forgotten Zagreb indices of linear [n]-Tetracene, V-Tetracenic nanotube, H-Tetracenic nanotube and Tetracenic nanotori were computed for the first time. These indices can be used in QSAR/QSPR studies.