Investigation on Tri-hexagonal Boron Nanotube by Exploiting the Certain Topological Indices and Their M-polynomials

Abstract 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.


Introduction
Chemical graph theory is an important branch of graph theory as it contains discrete Adriatic indices. In this theory, a descriptor of a molecular graph G is a number related to the structure of G and is invariant under the automorphisms of the graph.
Recently, D. Vukicevic [24] revealed the set of 148 discrete Adriatic indices. They were analyzed on the testing sets provided by the International Academy of Mathematical Chemistry (IAMC) and it had been shown that they have good predictive properties in many cases. There was a vast research regarding various properties of these topological indices.
There are two oldest degree based topological indices, first and second zagreb indices were introduced more than thirty years ago by Gutman and Trinajstic [11] which are defined as where denotes the degree of a vertex u in G. Both the first zagreb index and the second zagreb index give greater weights to the inner vertices and edges, and smaller weights to the outer vertices and edges, which opposes intuitive reasoning.
For a simple connected graph G, the second modified zagreb index [18] is defined as Symmetric division deg index [8,9] is one of the discrete Adriatic indices that is good predictor of total surface area for polychlorobiphenyls. The symmetric division degree index of a connected graph G is defined as The Randi´c index [19] also the most popular, the most often applied and the most studied among all other topological indices. The Randi´c index denoted by R(G) and introduced by Milan Randi´c in 1975, is also one of the oldest topological index.
In 1998, working independently, Bollobas and Erdos [2] and Amic et. al., [1] proposed the generalised Randi´c index and has been studied extensively by both chemists and mathematicians. The ordinary Randi´c connectivity index has been extended to the general Randi´c connectivity index defined as The inverse sum index (ISI) is a significant predictor of total surface area for octane isomers these are introduced by Damir Vukicevic and Marija Gasperov [4].
Another variant of the Randi´c index is the harmonic index [21] defined as The M-polynomial [14] of graph G is defined as where ( ), ( , ≥ 1) be the number of edges = of such that ( , ) = ( , ).

Table 1. Derivation of degree-based topological indices from M-polynomials
Symmetric division degree , ,

Tri-hexagonal Boron Nanotube
In 1991 nanotubes were discovery, it has created intense experimental and theoretical interest in such structures. Theoretical perspectives of carbon tubes suggest that their electrical properties will range from metallic to semiconducting, depending on the tube diameter.
In 2004 the first boron triangular nanotubes were innovated and evolved from a triangular sheet. The latest discovery of boron triangular nanotubes challenges the monopoly of carbon nanotubes (CNTs). Researchers believe that carbon nanotubes are less significant than boron triangular nanotubes. Eventually scientists have proved this speculation by innovating the world's smallest superconductor using nanoscale molecular superconducting boron wires [3]. As for as special features are concerned boron nanotubes also have some better features when compared to CNTs such as high chemical stability, high resistance to oxidation at high temperatures and are a stable wide band-gap semiconductor. Because of these properties, they can be used for applications at high temperatures or in corrosive environments such as batteries, fuel cells, super capacitors, high speed machines as solid lubricant.

Main Results
In this section, we present our main results. In the following, we determine the topological indices and M-polynomials of the tri-hexagonal boron nanotube. The representatives of these partite sets are shown in Figure 3 in which red, green, black and blue edges are edges belong to E (3;5) , E (4;4) , E (4;5) and E (5;5) respectively. Now consider, Similarly, the proof is followed by the same technique by taking into account of edge partition and then applying Equations

Conclusion
In this paper, we computed the certain degree-based topological indices and M-polynomial. Also, plotted the 3D structure for tri-hexagonal boron nanotube. The topological indices calculated in here can help us to understand the physical features, chemical reactivity, and biological activities. In this perspective, topological index is considered as vital role that maps each molecular structure to a real number and is used as a descriptor of the molecule under testing.