Abstract
Octa Graphene is a carbon nanosheet constructed by octagons and squares to build a unit cell. Topological properties of nano sheets based on Octa graphene are investigated in this article for the first time. Octa Graphene play a role as a Nano carrier for drug delivery and plays vital role in cell base drug delivery applications. Two structures, Namely Octa-Grayphyne and Octa-Graphdine are investigated topically in this article. Mechanically Octa-Grayphyne and Octa-Graphdine are of immense interest in Nano structural batteries and other Nano devices. The ultimate value of these Nano sheets in nano electronics can be interpreted by observing their electronic structure and vibrational properties.
1. Introduction
Nano sheets are two dimensional Nano structures. Carbon Nano sheets from the last decades are of great interest for researchers. These sheets were produced by heating the fibers at a temperature of over 350 F for 24 hours. Graphene is crystal like Nano structure of carbon atoms, with a strong bond and of thickness of one atom. Whereas Graphene Nano sheet is a 2D Nano structure consist of a single lyre of carbon atoms with hexagonal lattices. Wu et al. [1] investigated Graphene oxide sheet can be used as a carrier for Adriamcine, can reverse drug resistance in breast cancer. Moreover, in [2] it was shown that Graphene oxide sheet was also an anti-cancer agent. Without loss of generality we can state that Grephene oxide sheet is a source to decrease death rate due to cancer. The novel 2D Nano sheets have open new horizons to solve the serious environmental water pollution problem of the time by serving as a favorable candidate to reduce heavy metal pollution from water. To read about Nanosheets read [3].
Octa Grephene is a carbon Nano sheet, consisting of octagons and squares. Poupitz at el [4] introduces Octa-Grayphyne and Octa-Graphdine Nano sheets. Before going to further investigation of theses nano sheets, authors would like to recall some history of Topological indices for chemical graphs;
Let Ǥ be a graph with vertex set V(Ǥ) and edge set E(Ǥ) with and respectively. Degree of a vertex , means number of edges incident with and symbolically written as . A chemical Graph is the graphical representation of a chemical structure where vertices represents the atoms and edges represents the bond between atoms. In chemical graphs the maximum degree of vertex that can have is 4.
The Molecular Descriptor is a procedure to transform chemical information of a compound to a numerical value. Topological indices are one of the types of molecular descriptors. These indices are numerical values calculated from graphical representation of chemical compounds. Topological indices play an important role for Quantitative Structure Activity relationship (QSAR) by correlating psychochemical and biological properties of chemical compounds. Harry wiener is the founder of chemical graph theory. His contribution towards study of Topological indices is radical and about QSAR methods can be found in [5]commendable. He introduced structure-base invariants correlating boiling points of paraffin. He introduced the Winer index, which plays a major role in nanotechnology.
One of the classical topological index was introduces by Milan Randic in 1975. Randic index is the Sum of bond contribution of the degrees of vertices (atoms) making that bond, defined in [6] as;
Idea of general randic index was given by Bollobás and Erdös in [7] as:
Gutman and Trinajsti´c [8], introduces first and second zegreb indices and as;
History of topological index will not complete without discussion of Atom Bond connectivity index(ABC index) as it has extraordinary applications in rationalizing alkanes. ABC index was introduced by Estrada et al. [9] as;
Harmonic index of a graph is defined [10] as;
For a molecular graph the forgotten index or index [11] is;
Symmetric Division index or for a molecular Graph is defined [12] as;
The reader is encouraged to study [7, 13, 14] for more information on degree-based topological indices.
Deutsch, E.; Deutsch, [15] gave the concept of M polynomial of a chemical graph defined as;where denotes the number of the edges of molecular graph G such that ().
For further studies about M-polynomial please read [16–28].
Table 1gives the relation between the degree base topological indices with M polynomial.Where,
2. Main Results
2.1 Let be the Octa-Graphyne Nano sheet with m hexagons with each row and n hexagons in each column. From the graphical representation of Octa- Graphyne nano sheet, we can observe that each row of consists of m hexagons and m + 1 squares. The Graph of can be found in Figure 1. Each successive column of also consists n hexagons and n + 1 squares. Degrees of end vertices for all edges are used to calculate degree-based topological indices. has number of vertices, whereas number of edges for is . By observing graph of one can find that that all vertices of has either degree 2 or 3. Depending on the degree of vertices edge partition of is given in Table 2.
Partition of edge sets is defined as,
Based on edge partition given in Table 2, we can find M polynomial of Octa-Graphyne Nanosheet , Whereas Graphical representation of M polynomial of is given in Figure 2.
Theorem 1. Let be the graph of Octa- Graphyne Nanosheet, M polynomial for is given by,
Proof. Let be the graph of Octa- Graphyne Nanosheet, By using Table 2, combined with equation9, we get
Theorem 2. Let be the graph of Octa- Graphyne Nanosheet Let be M polynomial of then we have,
Proof. Let be M polynomial of Octa- Graphyne Nanosheet then,
Theorem 3. Let be the graph of Octa- Graphyne Nanosheet Let(i) be M polynomial of then we have,(ii)(iii)(iv)
Proof. Let be M polynomial of Octa- Graphyne Nanosheet then ,2.2. Let be the Octa-Graypdiyne Nanosheet with m)hexadecagones with each row and n hexadecagones in each column. From a graphical representation of the Octa-Grayphdine Nano sheet we can observe that each row of consists of m hexadecagones and m + 1 squares. The graph of Octa-Graypdiyne Nanosheet is given in Figure 3. Each successive column of also consists n hexadecagones and n + 1 squares. Degrees of end vertices for all edges are used to calculate degree-based topological indices. has number of vertices, whereas number of edges for is . By observing graph of one can find that that all vertices of has either degree 2 or 3. Depending on the degree of vertices edge partition of is given in Table 3.
Partition of edge sets is defined as,Based on edge partition given in Table 3, we can find M polynomial of Octa-Grayphdine Nanosheet , Whereas Graphical representation of M polynomial of is given in Figure 4.
Theorem 4. Let be the graph of Octa-Grayphdine Nanosheet, M polynomial for is given by,
Proof. Let be the graph of Octa-Grayphdine Nanosheet, Using Table 3, combined with equation9, we get
Theorem 5. Let be the graph of Octa-Grayphdine Nanosheet Let
be M polynomial of then we have(i)(ii)(iii)(iv)
Proof. Let be M polynomial of Octa-Grayphdine Nanosheet then,
Theorem 6. Let be the graph of Octa-Grayphdine Nanosheet Let
be M polynomial of then we have(i)(ii)(iii)
Proof. Let be M polynomial of Octa-Grayphdine Nanosheet then,
3. Conclusion
Various forms of degree-based topological indices, such as Randic index variations of Zagreb indices, atom bond connectivity index, Harmonicindex, and Forgotton index and Symmitric Division index are computed using M polynomia for nanosheets based on octa grephene. These discoveries are particularly useful in the theoretical research of nanosheet physical properties, chemical reactivity, and biological activities. The topological indices calculated in this study can be used in quantitative structure activity relations and quantitative structure property relations of nanosheets, which can assist further comprehend the nanosheet’s physiochemical properties.
Data Availability
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Conflicts of Interest
The authors declare that they have no conflicts of interest.