# Topological Properties of Nano Sheets Based on Octa Graphene

Authors: Fozia Bashir Farooq
Journal: Journal of Mathematics
 (2022)

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## 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.


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## Body

## 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) withVG=p and EG=q respectively. Degree of a vertex u∈VG, means number of edges incident with u and symbolically written as du. 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;(1)χǤ=∑uv∈EǤ1dudv.Idea of general randic index was given by Bollobás and Erdös in [7] as:(2)RαǤ=∑uv∈EǤdudvα.Gutman and Trinajsti´c [8], introduces first and second zegreb indices M1Ǥ and M2Ǥ as;(3)M1Ǥ=∑uv∈EǤdu+dv,M2Ǥ=∑uv∈EǤdudv.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;(4)IǤ=∑uv∈EǤdu+dv−2dudv.Harmonic indexHǤ of a graph Ǥ is defined [10] as;(5)HǤ=∑uv∈EǤ2du+dv.For a molecular graphǤ the forgotten index or F index [11] is;(6)FǤ=∑uv∈EǤdu2+dv2.Symmetric Division index orSDD for a molecular Graph Ǥ is defined [12] as;(7)SDDǤ=∑uv∈EǤmindu,dvmaxdu,dv+maxdu,dvmindu,dv.The reader is encouraged to study [7, 13, 14] for more information on degree-based topological indices.Deutsch, E.; Deutsch, [15] gave the cmindu,dv/maxdu,dv+maxdu,dv/mindu,dvoncept of M polynomial of a chemical graph defined as;(8)MǤ,x,y=∑i≤jeijǤαiβj,where eijǤ,i,j≥1 denotes the number of the edges e=uv of molecular graph G such that (du,dv=i, j).For further studies about M-polynomial please read [16–28].Table1gives the relation between the degree base topological indices with M polynomial.Where,(9)Dαfα,β=α·∂fα,β∂α,Dβfα,β=β·∂fα,β∂β,Jf=fα,α,Sαfα,β=∫0αft,βtdt,Sβfα,β=∫0βfα,ttdt.Table 1
Derivation of topological indices from theM polynomial.
Degree base topological indicesfα,βDerivation fromMǤ,α,βM1Ǥα+βDα+DβMǤ,α,β|α=β=1M2ǤαβDαDβMǤ,α,β|α=β=1R∝Ǥαβ∝∝∈NDα∝Dβ∝MǤ,α,β|α=β=1IǤαβ/α+β2SαJDαDβMǤ,α,β|α=β=1HǤ2/α+β2SαJMǤ,α,β|α=β=1FǤα2+β2Dα2+Dβ2MǤ,α,β│α=β=1SDDǤα2+β2/αβDαSβ+SαDβMǤ,α,β|α=β=1


## 2. Main Results

2.1 LetO1Ǥm×n 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 O1Ǥm×nconsists of m hexagons and m + 1 squares. The Graph of O1Ǥm×n can be found in Figure 1. Each successive column of O1Ǥm×n also consists n hexagons and n + 1 squares. Degrees of end vertices for all edges are used to calculate degree-based topological indices. O1Ǥm×n has 4mn+2m+2n number of vertices, whereas number of edges for O1Ǥm×n is 6mn+m+n. By observing graph of O1Ǥm×n one can find that that all vertices of O1Ǥm×n has either degree 2 or 3. Depending on the degree of vertices edge partition of O1Ǥm×n is given in Table 2.Figure 1
Graph of octa-graphyne nanosheetO1Ǥ3×3.Table 2
Edge Partition ofO1Ǥm×n Nano sheet.
Edge typesdu,dvFrequencyE2,22,22m+n+2E2,32,34m+n−2E3,33,36mn−5m−5n+4Partition of edge sets is defined as,(10)Ei,j=uv∈EO1Ǥm×n:du=ianddv=j,2≤i≤j≤3.Based on edge partition given in Table2, we can find M polynomial of Octa-Graphyne Nanosheet O1Ǥm×n , Whereas Graphical representation of M polynomial of O1Ǥm×n is given in Figure 2.Figure 2
The graph of the M-polynomial ofO1Ǥm×n Nanosheet.Theorem 1.
LetO1Ǥm×n be the graph of Octa- Graphyne Nanosheet, M polynomial for O1Ǥm×n is given by,(11)MO1Ǥm×n=2m+n+2α2β2+4m+n−2α2β3+6mn−5m−5n+4α3β3.Proof.
LetO1Ǥm×n be the graph of Octa- Graphyne Nanosheet, By using Table 2, combined with equation9, we get(12)MO1Ǥm×n=∑i≤jeijO1Ǥm×nαiβj=∑​e22O1Ǥm×nα2β2+∑​e23O1Ǥm×nα2β3+∑​e33O1Ǥm×nα3β3=2m+n+2α2β2+4m+n−2α2β3+6mn−5m−5n+4α3β3.Theorem 2.
LetO1Ǥm×n be the graph of Octa- Graphyne Nanosheet LetMO1Ǥm×n=2m+n+2α2β2+4m+n−2α2β3+6mn−5m−5n+4α3β3 be M polynomial of O1Ǥm×n then we have,(13)M1O1Ǥm×n=36mn−2m−2n,M2O1Ǥm×n=54mn−13m−13n+4,R∝O1Ǥm×n=232∝+1mn+22∝+1+2∝+23∝−5Ǥ32∝m+22∝+1+2∝+23∝−5Ǥ32∝n+422∝+32∝−2Ǥ2∝Ǥ3∝,IO1Ǥm×n=9mn−0⋅7m−0⋅7n+0⋅4.Proof.
LetMO1Ǥm×n,α,β=2m+n+2α2β2+4m+n−2α2β3+6mn−5m−5n+4α3β3 be M polynomial of Octa- Graphyne Nanosheet then, (14)iM1O1Ǥm×n=Dα+DβMO1Ǥ,α,β|α=β=1,DαMO1Ǥm×n,α,β=4m+n+2α2β2+8m+n−2α2β3+36mn−5m−5n+4α3β3,DβMO1Ǥm×n,α,β=4m+n+2α2β2+12m+n−2α2β3+36mn−5m−5n+4α3β3,Dα+DβMO1Ǥm×n,α,β=8m+n+2α2β2+20m+n−2α2β3+66mn−5m−5n+4α3β3,Dα+DβMO1Ǥ,α,β|α=β=1=8m+n+2+20m+n−2+66mn−5m−5n+4=36mn−2m−2n.iiM2O1Ǥm×n=DαDβMO1Ǥm×n,α,β|α=β=1,DαDβMO1Ǥm×n,α,β=8m+n+2α2β2+24m+n−2α2β3+96mn−5m−5n+4α3β3,DαDβMO1Ǥm×n,α,β|α=β=1=8m+n+2+24m+n−2+96mn−5m−5n+4=54mn−13m−13n+4.iiiR∝O1Ǥm×n=Dα∝Dβ∝MO1Ǥm×n,α,β|α=β=1,Dα∝Dβ∝MO1Ǥm×n,α,β=2Ǥ2∝2βm+n+2α2β2+4Ǥ2∝3∝m+n−2α2β3+3∝3∝6mn−2∝Ǥ5m−5n+4α3β3,Dα∝Dβ∝MO1Ǥm×n,α,β|α=β=1=2Ǥ2∝2∝m+n+2α2β2+4Ǥ2∝3∝m+n−2α2β3+3∝3∝6mn−5m−5n+4α3β3=232∝+1mn+22∝+1+2∝+23∝−5Ǥ32∝m+22∝+1+2∝+23∝−5Ǥ32∝n+422∝+32∝.ivIO1Ǥm×n=SαJDαDβMO1Ǥm×n,α|α=β=1,JDαDβMO1Ǥm×n,α=8m+n+2α4+24m+n−2α5+96mn−5m−5n+4α6,SαJDαDβMO1Ǥm×n,α=2m+n+2α4+245m+n−2α5+326mn−5m−5n+4α6,SαJDαDβMO1Ǥm×n,α|α=β=1=2m+n+2+245m+n−2+326mn−5m−5n+4,IO1Ǥm×n=9mn−0⋅7m−0⋅7n+0⋅4.Theorem 3.
LetO1Ǥm×n be the graph of Octa- Graphyne Nanosheet Let(i)
MO1Ǥm×n=2m+n+2α2β2+4m+n−2α2β3+6mn−5m−5n+4α3β3 be M polynomial of O1Ǥm×n then we have,(ii)
HO1Ǥm×n=2mn+0⋅933m+0⋅933n+0⋅133(iii)
FO1Ǥm×n=108mn−22m−22n(iv)
SDDO1Ǥm×n=12mn+2⋅67m+2⋅67n−1⋅33Proof.
LetMO1Ǥm×n,α,β=2m+n+2α2β2+4m+n−2α2β3+6mn−5m−5n+4α3β3 be M polynomial of Octa- Graphyne Nanosheet then ,(15)iHO1Ǥm×n=2SαJMO1Ǥm×n,α,β|α=β=1,JMO1Ǥm×n,α,β=2m+n+2α4+4m+n−2α5+6mn−5m−5n+4α6,2SαJMO1Ǥm×n,α,β=2mn+m+n+4α4+854mn+6m+6nα5+13mnα6,HO1Ǥm×n=2mn+0⋅933m+0⋅933n+0⋅133.iiFO1Ǥm×n=Dα2+Dβ2MO1Ǥm×n,α,β|α=β=1,Dα2MO1Ǥm×n,α,β=8m+n+2α2β2+16m+n−2α2β3+96mn−5m−5n+4α3β3,Dβ2MO1Ǥm×n,α,β=8m+n+2α2β2+36m+n−2α2β3+96mn−5m−5n+4α3β3,FO1Ǥm×n=108mn−22m−22n.iiiSDDO1Ǥm×n=DαSβ+SαDβMO1Ǥm×n,α,β|α=β=1,DαSβMO1Ǥm×n,α,β=2m+n+2α2β2+83m+n−2α2β3+136mn−5m−5n+4α3β3,DαSβMO1Ǥm×n,α,β|α=β=1=2m+n+2+83m+n−2+136mn−5m−5n+4,SαDβMO1Ǥm×n,α,β=2m+n+2α2β2+6m+n−2α2β3+6mn−5m−5n+4α3β3,SαDβMO1Ǥm×n,α,β|α=β=1=2m+n+2+6m+n−2+6mn−5m−5n+4,SDDO1Ǥm×n=12mn+2⋅67m+2⋅67n−1.33.
2.2. LetO2Ǥm×n 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 O2Ǥm×n consists of m hexadecagones and m + 1 squares. The graph of Octa-Graypdiyne Nanosheet O2Ǥm×n is given in Figure 3. Each successive column of O2Ǥm×nalso consists n hexadecagones and n + 1 squares. Degrees of end vertices for all edges are used to calculate degree-based topological indices. O2Ǥm×n has 8mn+6m+6n+4 number of vertices, whereas number of edges for O2Ǥm×n is 10mn+7m+7n+4. By observing graph of O2Ǥm×n one can find that that all vertices of O2Ǥm×nhas either degree 2 or 3. Depending on the degree of vertices edge partition of O2Ǥm×nis given in Table 3.
Partition of edge sets is defined as,(16)Ei,j=uv∈EO2Ǥm×n:du=ianddv=j2≤i≤j≤3.
Based on edge partition given in Table3, we can find M polynomial of Octa-Grayphdine Nanosheet O2Ǥm×n, Whereas Graphical representation of M polynomial of O2Ǥm×n is given in Figure 4.Figure 3
Graph of octa-graphdine nanosheetO2Ǥ3×3.Table 3
Edge Partition ofO2Ǥm×nNano sheet.
Edge typesdu,dvFrequencyE2,22,22mn+m+n+4E2,32,34mn+6m+6nE3,33,34mnFigure 4
The graph of M-polynomial ofO2Ǥm×n Nanosheet.Theorem 4.
LetO2Ǥm×n be the graph of Octa-Grayphdine Nanosheet, M polynomial for O2Ǥm×n is given by,(17)MO2Ǥm×n=2mn+m+n+4α2β2+4mn+6m+6nα2β3+4mnα3β3.Proof.
LetO2Ǥm×n be the graph of Octa-Grayphdine Nanosheet, Using Table 3, combined with equation9, we get(18)MO2Ǥm×n=∑i≤jeijO2Ǥm×nαiβj=∑e22O2Ǥm×nα2β2​+∑e23O2Ǥm×nα2β3+∑e33O2Ǥm×nα3β3​=2mn+m+n+4α2β2+4mn+6m+6nα2β3+4mnα3β3.Theorem 5.
LetO2Ǥm×n be the graph of Octa-Grayphdine Nanosheet Let
MO2Ǥm×n=2mn+m+n+4α2β2+4mn+6m+6nα2β3+4mnα3β3 be M polynomial of O2Ǥm×n then we have(i)
M1O2Ǥm×n=52mn+34m+34n+16.(ii)
M2O2Ǥm×n=68mn+40m+40n+16.(iii)
R∝O2Ǥm×n=24∝+2Ǥ6∝+2Ǥ9∝mn+4∝+6∝+1m+4∝+6∝+1n+4∝+1.(iv)
IO2Ǥm×n=12⋅8mn+8⋅2m+8⋅2n+4.Proof.
LetMO2Ǥm×n,α,β=2mn+m+n+4α2β2+4mn+6m+6nα2β3+4mnα3β3 be M polynomial of Octa-Grayphdine Nanosheet then, (19)iM1O2Ǥm×n=Dα+DβMO2Ǥm×n,α,β|α=β=1,DαMO2Ǥm×n,α,β=22mn+m+n+4α2β2+24mn+6m+6nα2β3+12mnα3β3,DβMO2Ǥm×n,α,β=22mn+m+n+4α2β2+34mn+6m+6nα2β3+12mnα3β3,Dα+DβMO2Ǥm×n,α,β=42mn+m+n+4α2β2+54mn+6m+6nα2β3+24mnα3β3Dα+DβMO2Ǥm×n,α,β│α=β=1=42mn+m+n+4+54mn+6m+6n+24mn=52mn+34m+34n+16.iiM2O2Ǥm×n=DαDβMO2Ǥm×n,α,β|α=β=1,DαDβMO2Ǥm×n,α,β=42mn+m+n+4α2β2+64mn+6m+6nα2β3+36mnα3β3,DαDβMO2Ǥm×n,α,β|α=β=1=42mn+m+n+4+64mn+6m+6n+36mn=68mn+40m+40n+16.iiiR∝O2Ǥm×n=Dα∝Dβ∝MO2Ǥm×n,α,β|α=β=1,Dα∝Dβ∝MO2Ǥm×n,α,β=2∝2∝2mn+m+n+4α2β2+2∝3∝4mn+6m+6nα2β3+4×3∝3∝mnα3β3,Dβ∝MO2Ǥm×n,α,β|α=β=1=2∝2∝2mn+m+n+4+2∝3∝4mn+6m+6n+4×3∝3∝mn,=24∝+2×6∝+2×9∝mn+4∝+6∝+1m+4∝+6∝+1n+4∝+1.IO2Ǥm×n=SαJDαDβMO2Ǥm×n,α,β|α=β=1,JDαDβMO2Ǥm×n,α=42mn+m+n+4α4+64mn+6m+6nα5+36mnα6,SαJDαDβMO2Ǥm×n,α,β=2mn+m+n+4α4+654mn+6m+6nα5+6mnα6,SαJDαDβMO2Ǥm×n,α,β|α=1=2mn+m+n+4+654mn+6m+6n+6mn,IO2Ǥm×n=64mn+41m+41n5+4.Theorem 6.
LetO2Ǥm×n be the graph of Octa-Grayphdine Nanosheet Let
MO2Ǥm×n=2mn+m+n+4α2β2+4mn+6m+6nα2β3+4mnα3β3 be M polynomial of O2Ǥm×n then we have(i)
HO2Ǥm×n=3.933mn+2.9m+2.9n+2.(ii)
FO2Ǥm×n=140mn+86m+86n+32.(iii)
SDDO2Ǥm×n=20.67mn+15m+15n+8.Proof.
LetMO2Ǥm×n,α,β=2mn+m+n+4α2β2+4mn+6m+6nα2β3+4mnα3β3 be M polynomial of Octa-Grayphdine Nanosheet then,(20)iHO2Ǥm×n=2SαJMO2Ǥm×n,α,β|α=β=1,JMO2Ǥm×n,α,β=2mn+m+n+4α4+4mn+6m+6nα5+4mnα6,SαJMO2Ǥm×n,α,β=142mn+m+n+4α4+154mn+6m+6nα5+23mnα6,HO2Ǥm×n=3⋅933mn+2⋅9m+2⋅9n+2.iiFO2Ǥm×n=Dα2+Dβ2MO2Ǥm×n,α,β|α=β=1,Dα2MO2Ǥm×n,α,β=42mn+m+n+4α2β2+44mn+6m+6nα2β3+36mnα3β3,Dβ2MO2Ǥm×n,α,β=42mn+m+n+4α2β2+94mn+6m+6nα2β3+36mnα3β3,FO2Ǥm×n=140mn+86m+86n+32.iiiSDDO2Ǥm×n=DαSβ+SαDβMO2Ǥm×n,α,β|α=β=1,DαSβMO2Ǥm×n,α,β|α=β=1=2mn+m+n+4+234mn+6m+6n+4mn,SαDβMO2Ǥm×n,α,β|α=β=1=2mn+m+n+4+324mn+6m+6n+4mn,SDDO2Ǥm×n=20.67mn+15m+15n+8.


## 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.



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