# Study of Graphene Networks and Line Graph of Graphene Networks viaNM-Polynomial and Topological Indices

Authors: Jie Wu; Saima Nazeer; Iftikhar Ahmed; Farkhanda Yasmin
Journal: Journal of Mathematics
 (2022)

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## Abstract

The topological invariants are related to the molecular graph of the chemical structure and are numerical numbers that help us to understand the topology of the concerned chemical structure. With the help of these numbers, many properties of graphene can be guessed without preforming any experiment. Huge amount of calculations are required to obtain topological invariants for graphene, but by applying basic calculus roles, neighborhoodM -polynomial of graphene gives its indices. The aim of this work is to compute neighborhood degree-dependent indices for the graph of graphene and the line graph of subdivision graph of graphene. Firstly, we establish neighborhood M-polynomial of these families of graphs, and then, by applying basic calculus, we obtain several neighborhood degree-dependent indices. Our results play an important role to understand graphene and enhance its abilities.


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## Body

## 1. Introduction

Graph is the union of edges (lines) and vertices (nodes), and the study of graphs is known as the graph theory [1, 2]. In this study, we study the properties of graphs and invent applications of different families of graphs [3, 4]. Today, the graph theory got applications in almost all areas of sciences and engineering, and applying the graph theory to solve problems of chemistry is known as the chemical graph theory [5, 6].In the chemical graph theory, topological indices are graph invariants that remain the same up to graph isomorphism and help us to attain properties of molecular graphs without any lab work. The first topological index was introduced by Wiener in 1947 when he was working on the boiling point of alkane [7]. Today, this index is named the Wiener index, which is a distance-based topological index. After Wiener, Randić introduced the first degree-depend index, which was firstly named the path index, but today, it is known as the Randić index [8]. This index got huge attention of researchers due to its amazing applications. After the promising success of the Randić index, Gutman coined the idea of Zagreb indices and many research papers are written on this index (see [9] and references therein).Since now, a single index can give all the information about any molecular graph, so till now, more than 150 indices are defined and studied [10–12]. Since huge amount of work is required to obtain topological indices, so the research found a shortcut to obtain them and introduced M-polynomial [13] and this polynomial gives easy way to compute almost all degree-based indices [14, 15]. Many studies have been performed so far on M-polynomial [16, 17], and still, there are many known interesting macular graphs whose M-polynomial can be established. Following M-polynomial, neighborhood M -polynomial was introduced in Ref. [18], which is helpful in determining neighborhood degree sum-based graphical indices.The aim of this work is to compute neighborhood degree-dependent indices for the graph of graphene and the line graph of subdivision graph of graphene. Firstly, we establish neighborhoodM-polynomial of these families of graphs, and then, by applying basic calculus, we obtain several neighborhood degree-dependent indices.


## 2. Primaries

Throughout this work, we consider only finite, simple, undirected, and connected graphs with a vertex setVG and an edge set EG [19, 20]. The degree dGv of a vertex v is the number of vertices adjacent to v [21, 22]. The neighborhood degree sum-based graphical indices depend upon the neighborhood degree, which is defined by dGe=dGu+dGv−2 for an edge e=uv, where u and v are vertices of the edge e. Neighborhood degree sum δu [23, 24] of a vertex u is defined as the sum of degrees of all vertices that are adjacent to the vertex. The degree of a vertex is the total number of edges incident to the vertex. The line graph LG of a graph G is the graph whose vertex set corresponds to the edges of G such that two vertices of LG are adjacent if the corresponding edges of G are adjacent (see for details [25, 26] and references therein). The subdivision graph SG of a graph G is the graph obtained from G by replacing each of its edges by a path of length two (see [27, 28]).The neighborhoodM -polynomial of a graph G is defined as [18](1)NMG;x,y=∑i≤jmi,jxiyj,where mi,j is the total number of edges uv∈EG such that δu,δv=i,j. We use NMG for NMG;x,y in this work.The neighborhood degree-based graphical indices defined on the edge set of a graphG can be expressed as(2)IG=∑uv∈EGfδu,δv,where fδu,δv is the function of δu,δv used in the definition of neighborhood degree-based indices. The above result can also be written as(3)IG=∑i≤jmi,jfi,j.By takingfδu,δv=δu+δv,δuδv,δu2+δv2,1/δuδv,δuδvα,δuδvδu+δv,δu/δv+δv/δu,2/δu+δv,δuδv/δu+δv,δuδv/δu+δv−23,δu+δv2,δuδv2,δu+δv/δuδv,2δuδv/δu+δv in equation (2), we get the third version of the Zagreb index, the neighborhood second Zagreb index, the neighborhood forgotten graphical index, the neighborhood second modified Zagreb index, the neighborhood general Randić index, the third NDe index, the fifth NDe index, the neighborhood Harmonic index, the neighborhood inverse sum index, the Sanskruti index, the fifth hyper M1 Zagreb index, the fifth hyper M2 Zagreb index, the fifth arithmetic-geometric index, and the fifth geometric-arithmetic index, respectively [29–32].All the abovementioned indices can be computed directly fromNM-polynomial, and the relations of some neighborhood degree-based graphical indices with the NM -polynomial are shown in Table 1, where Dxfx,y=x∂fx,y/∂x, Dyfx,y=y∂fx,y/∂y, Sxfx,y=∫0xft,y/tdt, Syfx,y=∫0yfx,t/tdt, and Jfx,y=fx,x, Qαfx,y=xαfx,y are the operators.Table 1
Derivation of some neighborhood degree-based graphical indices [18].
Graphical indexfx,yDerivation fromNMGM1′x+yDx+DyNMG|x=y=1M2∗xyDxDyNMG|x=y=1FN∗x2+y2Dx2+Dy2NMG|x=y=1M2nm1/xySxSyNMG|x=y=1NRαxyαDxαDyαNMG|x=y=1ND3xyx+yDxDyDx+DyNMG|x=y=1ND5x2+y2/xyDxSy+SxDyNMG|x=y=1NH2/x+y2SxJNMG|x=y=1NIxy/x+ySxJDxDyNMG|x=y=1Sxy/x+y−23Sx3Q−2JDx3Dy3NMG|x=y=1HM1G5Gx+y2Dx2+Dy2+2DxDyNMG|x=y=1HM2G5Gxy2DxDyDxDyNMG|x=y=1AG5Gx+y/2xy1/2Sx1/2Sy1/2Dx+DyNMG|x=y=1GA5G2xy/x+y2SxJDx1/2Dy1/2NMG|x=1


## 3. NeighborhoodM -Polynomial of Graphene Networks

In this section, we find neighborhoodM-polynomial of graphene and calculate neighborhood degree-based graphical indices of graphene by using its neighborhood M -polynomial.


### 3.1. Graphene Networks

Graphene is an atomic scale honeycomb lattice made of the carbon atoms. Graphene is denoted byGt,s, where t is the number of rows of benzene rings and s is the number of benzene rings in each row. Graphene has 2st+2s+2t vertices and 3st+2s+2t−1 edges [33]. Figure 1 shows the graphene Gt,s.Figure 1
GrapheneGt,s.Theorem 1.
LetGt,sbe a graphene network, witht>1ands>1; then, its neighborhoodM-polynomial is given by(4)NMGt,s=4x4y5+tx5y5+8x5y7+2t−4x5y8+4s−8x6y7+2sx7y9+t−2x8y8+2t−4x8y9+3st−4s−4t+5x9y9.Proof.
Graphene has2st+2s+2t vertices and 3st+2s+2t−1 edges. The edge set of Gt,s, for t>1 and s>1, can be partitioned as follows:(5)E4,5=uv∈EGt,s:δu=4,δv=5=4=m4,5,E5,5=uv∈EGt,s:δu=5,δv=5=t=m5,5,E5,7=uv∈EGt,s:δu=5,δv=7=8=m5,7,E5,8=uv∈EGt,s:δu=5,δv=8=2t−4=m5,8,E6,7=uv∈EGt,s:δu=6,δv=7=4s−8=m6,7,E7,9=uv∈EGt,s:δu=7,δv=9=2s=m7,9,E8,8=uv∈EGt,s:δu=8,δv=8=t−2=m8,8,E8,9=uv∈EGt,s:δu=8,δv=9=2t−4=m8,9,E9,9=uv∈EGt,s:δu=9,δv=9=3st−4s−4t+5=m9,9.
Thus, the neighborhoodM -polynomial of Gt,s, for t>1 and s>1, is(6)NMGt,s=∑i≤jmi,jxiyj,NMGt,s=4x4y5+tx5y5+8x5y7+2t−4x5y8+4s−8x6y7+2sx7y9+t−2x8y8+2t−4x8y9+3st−4s−4t+5x9y9,
This is the required neighborhoodM -polynomial of Gt,s, for t>1 and s>1.Theorem 2.
LetGt,sbe a graphene network, witht=1ands>1; then, its neighborhoodM-polynomial is given by(7)NMGt,s=2x4y4+4x4y5+4x5y7+4s−8x6y7+s−1x7y7.Proof.
Graphene has2st+2s+2t vertices and 3st+2s+2t−1 edges. The edge set of Gt,s, for t=1 and s>1, can be partitioned as follows:(8)E4,4=uv∈EGt,s:δu=4,δv=4=2=m4,4,E4,5=uv∈EGt,s:δu=4,δv=5=4=m4,5,E5,7=uv∈EGt,s:δu=5,δv=7=4=m5,7,E6,7=uv∈EGt,s:δu=6,δv=7=4s−8=m6,7,E7,7=uv∈EGt,s:δu=7,δv=7=s−1=m7,7.
Thus, the neighborhoodM -polynomial of Gt,s, for t=1 and s>1, is(9)NMGt,s=∑i≤jmi,jxiyj,NMGt,s=2x4y4+4x4y5+4x5y7+4s−8x6y7+s−1x7y7.
This is the required neighborhoodM -polynomial of Gt,s, for t=1 and s>1.Theorem 3.
LetGt,sbe a graphene network, witht>2ands=1; then, its neighborhoodM-polynomial is given by(10)NMGt,s=2x4y4+4x4y5+t−2x5y5+4x5y7+2t−4x5y8+2x7y8+2t−5x8y8.Proof.
Graphene has2st+2s+2t vertices and 3st+2s+2t−1 edges. The edge set of Gt,s, for t>2 and s=1, can be partitioned as follows:(11)E4,4=uv∈EGt,s:δu=4,δv=4=2=m4,4,E4,5=uv∈EGt,s:δu=4,δv=5=4=m4,5,E5,5=uv∈EGt,s:δu=5,δv=5=t−2=m5,5,E5,7=uv∈EGt,s:δu=5,δv=7=4=m5,7,E5,8=uv∈EGt,s:δu=5,δv=8=2t−4=m5,8,E7,8=uv∈EGt,s:δu=7,δv=8=2=m7,8,E8,8=uv∈EGt,s:δu=8,δv=8=2t−5=m8,8.
Thus, the neighborhoodM -polynomial of Gt,s, for t>2 and s=1, is(12)NMGt,s=∑i≤jmi,jxiyj,NMGt,s=2x4y4+4x4y5+t−2x5y5+4x5y7+2t−4x5y8+2x7y8+2t−5x8y8,this is the required neighborhood M -polynomial of Gt,s, for t>2 and s=1.
Now, we calculate neighborhood degree-based graphical indices of graphene by using its neighborhoodM -polynomial.Corollary 1.
LetGt,sbe a graphene network, witht>1ands>1;then, its neighborhood degree-based graphical indices are given by(1)
M1′=54st+12s+14t−34(2)
M2∗=243st−30s−11t−147(3)
FN∗=486st−48s−2t−306(4)
M2nm=1/813st−4s−4t+5+2/21s−2+269/2880t−2+2/63s+1/25t+3/7(5)
NRα=20α4+25αt+35α8+40α2t−2+42α4s−2+63α2s+64αt−2+72α2t−2+81α3st−4s−4t+5(6)
ND3=4374st−1632s−1070t−2022(7)
ND5=23st−4s−4t+5+170/21s−2+943/90t−2+260/63s+2t+879/35(8)
NH=1/93st−4s−4t+5+8/13s−2+1181/1768t−2+1/4s+1/5t+20/9(9)
NI=9/23st−4s−4t+5+168/13s−2+4116/221t−2+63/8s+5/2t+290/9(10)
S=129.74633793st−4s−4t+5+222.6536439s−2+412.8858222t−2+729/4s+15625/512t+436.2944606(11)
HM1G5Gt,s=972st−108s−24t−600(12)
HM2G5Gt,s=19683st−11250s−7955t−5235(13)
AG5Gt,s=3st−4s−4t+5+4.011887099s−2+5.058949692t−2+2.015810523s+t+12.1384032(14)
GA5Gt,s=3st−4s−4t+5+842/13s−2+4.942553816t−2+37/4s+t+11.86333834Proof.
NM -Polynomial of Gt,s, for t>1 and s>1, is given by(13)NMGt,s=4x4y5+tx5y5+8x5y7+2t−2x5y8+4s−2x6y7+2sx7y9+t−2x8y8+2t−2x8y9+3st−4s−4t+5x9y9.
Let(14)NMGt,s=fx,y.
Then, we have(15)fx,y=NMGt,s=4x4y5+tx5y5+8x5y7+2t−2x5y8+4s−2x6y7+2sx7y9+t−2x8y8+2t−2x8y9+3st−4s−4t+5x9y9.(1)
M1′ is defined as(16)M1′=Dx+DyNMG|x=y=1.Now, by using equation (15), we have(17)M1′=Dx+Dyfx,y|x=y=1,M1′=Dx+Dy4x4y5+tx5y5+8x5y7+2t−2x5y8+4s−2x6y7+2sx7y9+t−2x8y8+2t−2x8y9+3st−4s−4t+5x9y9|x=y=1,M1′=16x4y5+20x4y5+5tx5y5+5tx5y5+40x5y7+56x5y7+10t−2x5y8+16t−2x5y8+24s−2x6y7+28s−2x6y7+14sx7y9+18sx7y9+8t−2x8y8+8t−2x8y8+16t−2x8y9+18t−2x8y9+93st−4s−4t+5x9y9+93st−4s−4t+5x9y9|x=y=1,M1′=54st+12s+14t−34.(2)
M2∗ is defined as(18)M2∗=DxDyNMG|x=y=1.Now, by using equation (15), we have(19)M2∗=DxDyfx,y|x=y=1,M2∗=DxDy4x4y5+tx5y5+8x5y7+2t−2x5y8+4s−2x6y7+2sx7y9+t−2x8y8+2t−2x8y9+3st−4s−4t+5x9y9|x=y=1,M2∗=80x4y5+25tx5y5+280x5y7+80t−2x5y8+168s−2x6y7+126sx7y9+64t−2x8y8+144t−2x8y9+813st−4s−4t+5x9y9|x=y=1,M2∗=243st−30s−11t−147.(3)
FN∗ is defined as(20)FN∗=Dx2+Dy2NMG|x=y=1.Now, by using equation (15), we have(21)FN∗=Dx2+Dy2fx,y|x=y=1,FN∗=Dx2+Dy24x4y5+tx5y5+8x5y7+2t−2x5y8+4s−2x6y7+2sx7y9+t−2x8y8+2t−2x8y9+3st−4s−4t+5x9y9|x=y=1,FN∗=64x4y5+100x4y5+25tx5y5+25tx5y5+200x5y7+392x5y7+50t−2x5y8+128t−2x5y8+144s−2x6y7+196s−2x6y7+98sx7y9+162sx7y9+64t−2x8y8+64t−2x8y8+128t−2x8y9+162t−2x8y9+813st−4s−4t+5x9y9+813st−4s−4t+5x9y9|x=y=1,FN∗=486st−48s−2t−306.(4)
M2nm is defined as(22)M2nm=SxSyNMG|x=y=1.Now, by using equation (15), we have(23)M2nm=SxSyfx,y|x=y=1,M2nm=SxSy4x4y5+tx5y5+8x5y7+2t−2x5y8+4s−2x6y7+2sx7y9+t−2x8y8+2t−2x8y9+3st−4s−4t+5x9y9|x=y=1,M2nm=420x4y5+125tx5y5+835x5y7+240t−2x5y8+442s−2x6y7+263sx7y9+164t−2x8y8+272t−2x8y9+1813st−4s−4t+5x9y9|x=y=1,M2nm=1813st−4s−4t+5+221s−2+2692880t−2+263s+125t+37.(5)
NRα is defined as(24)NRα=DxαDyαNMG|x=y=1.Now, by using equation (15), we have(25)NRα=DxαDyαfx,y|x=y=1,NRα=DxαDyα4x4y5+tx5y5+8x5y7+2t−2x5y8+4s−2x6y7+2sx7y9+t−2x8y8+2t−2x8y9+3st−4s−4t+5x9y9|x=y=1,NRα=20α4x4y5+25αtx5y5+35α8x5y7+40α2t−2x5y8+42α4s−2x6y7+63α2sx7y9+64αt−2x8y8+72α2t−2x8y9+81α3st−4s−4t+5x9y9|x=y=1,NRα=20α4+25αt+35α8+40α2t−2+42α4s−2+63α2s+64αt−2+72α2t−2+81α3st−4s−4t+5.(6)
ND3 is defined as(26)ND3=DxDyDx+DyNMG|x=y=1.Now, by using equation (15), we have(27)ND3=DxDyDx+Dyfx,y|x=y=1,ND3=DxDyDx+Dy4x4y5+tx5y5+8x5y7+2t−2x5y8+4s−2x6y7+2sx7y9+t−2x8y8+2t−2x8y9+3st−4s−4t+5x9y9|x=y=1,ND3=DxDy16x4y5+20x4y5+5tx5y5+5tx5y5+40x5y7+56x5y7+10t−2x5y8+16t−2x5y8+24s−2x6y7+28s−2x6y7+14sx7y9+18sx7y9+8t−2x8y8+8t−2x8y8+16t−2x8y9+18t−2x8y9+93st−4s−4t+5x9y9+93st−4s−4t+5x9y9|x=y=1,ND3=720x4y5+250tx5y5+3360x5y7+1040t−2x5y8+2184s−2x6y7+2016sx7y9+1024t−2x8y8+2448t−2x8y9+14583st−4s−4t+5x9y9|x=y=1,ND3=4374st−1632s−1070t−2022.(7)
ND5 is defined as(28)ND5=DxSy+SxDyNMG|x=y=1.Now, by using equation (15), we have(29)ND5=DxSy+SxDyfx,y|x=y=1.ND5=DxSy+SxDy4x4y5+tx5y5+8x5y7+2t−2x5y8+4s−2x6y7+2sx7y9+t−2x8y8+2t−2x8y9+3st−4s−4t+5x9y9|x=y=1,ND5=165x4y5+204x4y5+tx5y5+tx5y5+407x5y7+565x5y7+108t−2x5y8+165t−2x5y8+247s−2x6y7+286s−2x6y7+149sx7y9+187sx7y9+t−2x8y8+t−2x8y8+169t−2x8y9+188t−2x8y9+3st−4s−4t+5x9y9+3st−4s−4t+5x9y9|x=y=1,ND5=23st−4s−4t+5+17021s−2+94390t−2+26063s+2t+87935.(8)
NH is defined as(30)NH=2SxJNMG|x=y=1.Now, by using equation (15), we have(31)NH=2SxJfx,y|x=y=1,NH=2SxJ4x4y5+tx5y5+8x5y7+2t−2x5y8+4s−2x6y7+2sx7y9+t−2x8y8+2t−2x8y9+3st−4s−4t+5x9y9|x=y=1,NH=89x9+210tx10+1612x12+413t−2x13+813s−2x13+416sx16+216t−2x16+417t−2x17+2183st−4s−4t+5x18|x=1,NH=193st−4s−4t+5+813s−2+11811768t−2+14s+15t+209.(9)
NI is defined as(32)NI=SxJDxDyNMG|x=y=1.Now, by using equation (15), we have(33)NI=SxJDxDyfx,y|x=y=1,NI=SxJDxDy4x4y5+tx5y5+8x5y7+2t−2x5y8+4s−2x6y7+2sx7y9+t−2x8y8+2t−2x8y9+3st−4s−4t+5x9y9|x=y=1,NI=SxJ80x4y5+25tx5y5+280x5y7+80t−2x5y8+168s−2x6y7+126sx7y9+64t−2x8y8+144t−2x8y9+813st−4s−4t+5x9y9|x=y=1,NI=809x9+2510tx10+28012x12+8013t−2x13+16813s−2x13+12616sx16+6416t−2x16+14417t−2x17+81183st−4s−4t+5x18|x=1,NI=923st−4s−4t+5+16813s−2+4116221t−2+638s+52t+2909.(10)
S is defined as(34)S=Sx3Q−2JDx3Dy3NMG|x=y=1.Now, by using equation (15), we have(35)S=Sx3Q−2JDx3Dy3fx,y|x=y=1,S=Sx3Q−2JDx3Dy34x4y5+tx5y5+8x5y7+2t−2x5y8+4s−2x6y7+2sx7y9+t−2x8y8+2t−2x8y9+3st−4s−4t+5x9y9|x=y=1,S=32000343x7+15625512tx8+3430001000x10+1280001331t−2x11+2963521331s−2x11+5000942744sx14+2621442744t−2x14+7464963375t−2x15+53144140963st−4s−4t+5x16|x=1,S=129.74633793st−4s−4t+5+222.6536439s−2+412.8858222t−2+7294s+15625512t+436.2944606.(11)
Fifth hyperM1 Zagreb index is defined as(36)HM1G5G=Dx2+Dy2+2DxDyNMG|x=y=1.Now, by using equation (15), we have(37)HM1G5Gt,s=Dx2+Dy2+2DxDyfx,y|x=y=1,HM1G5Gt,s=Dx2+Dy2+2DxDy4x4y5+tx5y5+8x5y7+2t−2x5y8+4s−2x6y7+2sx7y9+t−2x8y8+2t−2x8y9+3st−4s−4t+5x9y9|x=y=1,HM1G5Gt,s=64x4y5+100x4y5+25tx5y5+25tx5y5+200x5y7+392x5y7+50t−2x5y8+128t−2x5y8+144s−2x6y7+196s−2x6y7+98sx7y9+162sx7y9+64t−2x8y8+64t−2x8y8+128t−2x8y9+162t−2x8y9+813st−4s−4t+5x9y9+813st−4s−4t+5x9y9+280x4y5+25tx5y5+280x5y7+80t−2x5y8+168s−2x6y7+126sx7y9+64t−2x8y8+144t−2x8y9+813st−4s−4t+5x9y9|x=y=1,HM1G5Gt,s=972st−108s−24t−600.(12)
Fifth hyperM2 Zagreb index is defined as(38)HM2G5G=DxDyDxDyNMG|x=y=1.Now, by using equation (15), we have(39)HM2G5Gt,s=DxDyDxDyfx,y|x=y=1,HM2G5Gt,s=DxDyDxDy4x4y5+tx5y5+8x5y7+2t−2x5y8+4s−2x6y7+2sx7y9+t−2x8y8+2t−2x8y9+3st−4s−4t+5x9y9|x=y=1,HM2G5Gt,s=DxDy80x4y5+25tx5y5+280x5y7+80t−2x5y8+168s−2x6y7+126sx7y9+64t−2x8y8+144t−2x8y9+813st−4s−4t+5x9y9|x=y=1,HM2G5Gt,s=1600x4y5+625tx5y5+9800x5y7+3200t−2x5y8+7056s−2x6y7+7938sx7y9+4096t−2x8y8+10368t−2x8y9+65613st−4s−4t+5x9y9|x=y=1,HM2G5Gt,s=19683st−11250s−7955t−5235.(13)
Fifth arithmetic-geometric index is defined as(40)AG5G=12Sx1/2Sy1/2Dx+DyNMG|x=y=1.Now, by using equation (15), we have(41)AG5Gt,s=12Sx1/2Sy1/2Dx+Dyfx,y|x=y=1,AG5Gt,s=12Sx1/2Sy1/2Dx+Dy4x4y5+tx5y5+8x5y7+2t−2x5y8+4s−2x6y7+2sx7y9+t−2x8y8+2t−2x8y9+3st−4s−4t+5x9y9|x=y=1,AG5Gt,s=12Sx1/2Sy1/216x4y5+20x4y5+5tx5y5+5tx5y5+40x5y7+56x5y7+10t−2x5y8+16t−2x5y8+24s−2x6y7+28s−2x6y7+14sx7y9+18sx7y9+8t−2x8y8+8t−2x8y8+16t−2x8y9+18t−2x8y9+93st−4s−4t+5x9y9+93st−4s−4t+5x9y9|x=y=1,AG5Gt,s=36220x4y5+10225tx5y5+96235x5y7+26240t−2x5y8+52242s−2x6y7+32263sx7y9+16264t−2x8y8+34272t−2x8y9+182813st−4s−4t+5x9y9|x=y=1,AG5Gt,s=3st−4s−4t+5+4.011887099s−2+5.058949692t−2+2.015810523s+t+12.1384032.(14)
Fifth geometric-arithmetic index is defined as(42)GA5G=2SxJDx1/2Dy1/2NMG|x=1,Now, by using equation (15), we have(43)GA5Gt,s=2SxJDx1/2Dy1/2fx,y|x=1,GA5Gt,s=2SxJDx1/2Dy1/24x4y5+tx5y5+8x5y7+2t−2x5y8+4s−2x6y7+2sx7y9+t−2x8y8+2t−2x8y9+3st−4s−4t+5x9y9|x=1,GA5Gt,s=2SxJ420x4y5+25tx5y5+835x5y7+240t−2x5y8+442s−2x6y7+263sx7y9+64t−2x8y8+272t−2x8y9+813st−4s−4t+5x9y9|x=1,GA5Gt,s=8209x9+22510tx10+163512x12+44013t−2x13c+84213s−2x13+46316sx16+26416t−2x16+47217t−2x17+281183st−4s−4t+5x18|x=1,GA5Gt,s=3st−4s−4t+5+84213s−2+4.942553816t−2+374s+t+11.86333834.Corollary 2.
LetGt,sbe a graphene network, witht=1ands>1; then, its neighborhood degree-based graphical indices are given by(1)
M1′=66s−18(2)
M2∗=217s−133(3)
FN∗=438s−254(4)
M2nm=2/21s−2+1/49s−1+123/280(5)
NRα=16α2+20α4+35α4+42α4s−2+49αs−1(6)
ND3=2870s−2398(7)
ND5=170/21s−2+2s−1+723/35(8)
NH=8/13s−2+1/7s−1+37/18(9)
NI=168/13s−2+7/2s−1+221/9(10)
S=222.6536439s−2+68.08391204s−1+302.7203866(11)
HM1G5Gt,s=872s−520(12)
HM2G5Gt,s=9457s−9501(13)
AG5Gt,s=1342/21s−2+s−1+10.08166278(14)
GA5Gt,s=842/13s−2+s−1+9.919285149Proof.
NM -Polynomial of Gt,s, for t=1 and s>1, is given by(44)NMGt,s=2x4y4+4x4y5+4x5y7+4s−2x6y7+s−1x7y7.
Let,(45)NMGt,s=fx,y,then, we have(46)fx,y=NMGt,s=2x4y4+4x4y5+4x5y7+4s−2x6y7+s−1x7y7.(1)
M1′ is defined as(47)M1′=Dx+DyNMG|x=y=1.Now, by using equation (46), we have(48)M1′=Dx+Dyfx,y|x=y=1,M1′=Dx+Dy2x4y4+4x4y5+4x5y7+4s−2x6y7+s−1x7y7|x=y=1,M1′=8x4y4+8x4y4+16x4y5+20x4y5+20x5y7+28x5y7+24s−2x6y7+28s−2x6y7+7s−1x7y7+7s−1x7y7|x=y=1,M1′=66s−18.(2)
M2∗ is defined as(49)M2∗=DxDyNMG|x=y=1.Now, by using equation (46), we have(50)M2∗=DxDyfx,y|x=y=1,M2∗=DxDy2x4y4+4x4y5+4x5y7+4s−2x6y7+s−1x7y7|x=y=1,M2∗=32x4y4+80x4y5+140x5y7+168s−2x6y7+49s−1x7y7|x=y=1,M2∗=217s−133.(3)
FN∗ is defined as(51)FN∗=Dx2+Dy2NMG|x=y=1.Now, by using equation (46), we have(52)FN∗=Dx2+Dy2fx,y|x=y=1,FN∗=Dx2+Dy22x4y4+4x4y5+4x5y7+4s−2x6y7+s−1x7y7|x=y=1,FN∗=32x4y4+32x4y4+64x4y5+100x4y5+100x5y7+196x5y7+144s−2x6y7+196s−2x6y7+490s−1x7y7+49s−1x7y7|x=y=1,FN∗=438s−254.(4)
M2nm is defined as(53)M2nm=SxSyNMG|x=y=1.Now, by using equation (46), we have(54)M2nm=SxSyfx,y|x=y=1,M2nm=SxSy2x4y4+4x4y5+4x5y7+4s−2x6y7+s−1x7y7|x=y=1,M2nm=216x4y4+420x4y5+435x5y7+442s−2x6y7+149s−1x7y7|x=y=1,M2nm=221s−2+149s−1+123280.(5)
NRα is defined as(55)NRα=DxαDyαNMG|x=y=1.Now, by using equation (46), we have(56)NRα=DxαDyαfx,y|x=y=1,NRα=DxαDyα2x4y4+4x4y5+4x5y7+4s−2x6y7+s−1x7y7|x=y=1,NRα=16α2x4y4+20α4x4y5+35α4x5y7+42α4s−2x6y7+49αs−1x7y7|x=y=1,NRα=16α2+20α4+35α4+42α4s−2+49αs−1.(6)
ND3 is defined as(57)ND3=DxDyDx+DyNMG|x=y=1.Now, by using equation (46), we have(58)ND3=DxDyDx+Dyfx,y|x=y=1,ND3=DxDyDx+Dy2x4y4+4x4y5+4x5y7+4s−2x6y7+s−1x7y7|x=y=1,ND3=DxDy8x4y4+8x4y4+16x4y5+20x4y5+20x5y7+28x5y7+24s−2x6y7+28s−2x6y7+7s−1x7y7+7s−1x7y7|x=y=1,ND3=256x4y4+720x4y5+1680x5y7+2184s−2x6y7+686s−1x7y7|x=y=1,ND3=2870s−2398.(7)
ND5 is defined as(59)ND5=DxSy+SxDyNMG|x=y=1.Now, by using equation (46), we have(60)ND5=DxSy+SxDyfx,y|x=y=1,ND5=DxSy+SxDy2x4y4+4x4y5+4x5y7+4s−2x6y7+s−1x7y7|x=y=1,ND5=2x4y4+2x4y4+165x4y5+204x4y5+207x5y7+285x5y7+247s−2x6y7+286s−2x6y7+s−1x7y7+s−1x7y7|x=y=1,ND5=17021s−2+2s−1+72335.(8)
NH is defined as(61)NH=2SxJNMG|x=y=1.Now, by using equation (46), we have(62)NH=2SxJfx,y|x=y=1,NH=2SxJ2x4y4+4x4y5+4x5y7+4s−2x6y7+s−1x7y7|x=y=1,NH=48x8+89x9+812x12+813s−2x13+214s−1x14|x=1,NH=813s−2+17s−1+3718.(9)
NI is defined as(63)NI=SxJDxDyNMG|x=y=1.Now, by using equation (46), we have(64)NI=SxJDxDyfx,y|x=y=1,NI=SxJDxDy2x4y4+4x4y5+4x5y7+4s−2x6y7+s−1x7y7|x=y=1,NI=SxJ32x4y4+80x4y5+140x5y7+168s−2x6y7+49s−1x7y7|x=y=1,NI=328x8+809x9+14012x12+16813s−2x13+4914s−1x14|x=1,NI=16813s−2+72s−1+2219.(10)
S is defined as(65)S=Sx3Q−2JDx3Dy3NMG|x=y=1.Now, by using equation (46), we have(66)S=Sx3Q−2JDx3Dy3fx,y|x=y=1,S=Sx3Q−2JDx3Dy32x4y4+4x4y5+4x5y7+4s−2x6y7+s−1x7y7|x=y=1,S=8192216x6+32000343x7+1715001000x10+2963521331s−2x11+1176491728s−1x12|x=1,S=222.6536439s−2+68.08391204s−1+302.7203866.(11)
Fifth hyperM1 Zagreb index is defined as(67)HM1G5G=Dx2+Dy2+2DxDyNMG|x=y=1.Now, by using equation (46), we have(68)HM1G5Gt,s=Dx2+Dy2+2DxDyfx,y|x=y=1,HM1G5Gt,s=Dx2+Dy2+2DxDy2x4y4+4x4y5+4x5y7+4s−2x6y7+s−1x7y7|x=y=1,HM1G5Gt,s=32x4y4+32x4y4+64x4y5+100x4y5+100x5y7+196x5y7+144s−2x6y7+196s−2x6y7+49s−1x7y7+49s−1x7y7+232x4y4+80x4y5+140x5y7+168s−2x6y7+49s−1x7y7|x=y=1,HM1G5Gt,s=872s−520.(12)
Fifth hyperM2 Zagreb index is defined as(69)HM2G5G=DxDyDxDyNMG|x=y=1.Now, by using equation (46), we have(70)HM2G5Gt,s=DxDyDxDyfx,y|x=y=1,HM2G5Gt,s=DxDyDxDy2x4y4+4x4y5+4x5y7+4s−2x6y7+s−1x7y7|x=y=1,HM2G5Gt,s=DxDy32x4y4+80x4y5+140x5y7+168s−2x6y7+49s−1x7y7|x=y=1,HM2G5Gt,s=512x4y4+1600x4y5+4900x5y7+7056s−2x6y7+2401s−1x7y7|x=y=1,HM2G5Gt,s=9457s−9501.(13)
Fifth arithmetic-geometric index is defined as(71)AG5G=12Sx1/2Sy1/2Dx+DyNMG|x=y=1.Now, by using equation (46), we have(72)AG5Gt,s=12Sx1/2Sy1/2Dx+Dyfx,y|x=y=1,AG5Gt,s=12Sx1/2Sy1/2Dx+Dy2x4y4+4x4y5+4x5y7+4s−2x6y7+s−1x7y7|x=y=1,AG5Gt,s=12Sx1/2Sy1/28x4y4+8x4y4+16x4y5+20x4y5+20x5y7+28x5y7+24s−2x6y7+28s−2x6y7+7s−1x7y7+7s−1x7y7|x=y=1,AG5Gt,s=16216x4y4+36220x4y5+48235x5y7+52242s−2x6y7+14249s−1x7y7|x=y=1,AG5Gt,s=134221s−2+s−1+10.08166278.(14)
Fifth geometric-arithmetic index is defined as(73)GA5G=2SxJDx1/2Dy1/2NMG|x=1.Now, by using equation (46), we have(74)GA5Gt,s=2SxJDx1/2Dy1/2fx,y|x=1,GA5Gt,s=2SxJDx1/2Dy1/22x4y4+4x4y5+4x5y7+4s−2x6y7+s−1x7y7|x=1,GA5Gt,s=4168x8+8209x9+83512x12+84213s−2x13+24914s−1x14|x=1,GA5Gt,s=84213s−2+s−1+9.919285149.Corollary 3.
LetGt,sbe a graphene network, witht>2ands=1; then, its neighborhood degree-based graphical indices are given by(1)
M1′=68t−22(2)
M2∗=233t−166(3)
FN∗=484t−346(4)
M2nm=1/642t−5+9/100t−2+19/40(5)
NRα=16α2+20α4+25αt−2+35α4+40α2t−2+56α2+64α2t−5(6)
ND3=3338t−3364(7)
ND5=22t−5+129/20t−2+3457/140(8)
NH=1/82t−5+33/65t−2+209/90(9)
NI=42t−5+225/26t−2+1441/45(10)
S=95.53352772t−5+126.6858726t−2+462.5892987(11)
HM1G5Gt,s=950t−678(12)
HM2G5Gt,s=12017t−14846(13)
AG5Gt,s=2t−5+3.055480479t−2+12.0861221(14)
GA5Gt,s=2t−5+2.946017022t−2+11.91483576Proof.
NM -Polynomial of Gt,s, for t>2 and s=1, is given by(75)NMGt,s=2x4y4+4x4y5+t−2x5y5+4x5y7+2t−2x5y8+2x7y8+2t−5x8y8.
Let,(76)NMGt,s=fx,y.
Then, we have(77)fx,y=NMGt,s=2x4y4+4x4y5+t−2x5y5+4x5y7+2t−2x5y8+2x7y8+2t−5x8y8.(1)
M1′ is defined as(78)M1′=Dx+DyNMG|x=y=1.Now, by using equation (77), we have(79)M1′=Dx+Dyfx,y|x=y=1,M1′=Dx+Dy2x4y4+4x4y5+t−2x5y5+4x5y7+2t−2x5y8+2x7y8+2t−5x8y8|x=y=1,M1′=8x4y4+8x4y4+16x4y5+20x4y5+5t−2x5y5+5t−2x5y5+20x5y7+28x5y7+10t−2x5y8+16t−2x5y8+14x7y8+16x7y8+82t−5x8y8+82t−5x8y8|x=y=1,M1′=68t−22.(2)
M2∗ is defined as(80)M2∗=DxDyNMG|x=y=1.Now, by using equation (77), we have(81)M2∗=DxDyfx,y|x=y=1,M2∗=DxDy2x4y4+4x4y5+t−2x5y5+4x5y7+2t−2x5y8+2x7y8+2t−5x8y8|x=y=1,M2∗=32x4y4+80x4y5+25t−2x5y5+140x5y7+80t−2x5y8+112x7y8+642t−5x8y8|x=y=1,M2∗=233t−166.(3)
FN∗ is defined as(82)FN∗=Dx2+Dy2NMG|x=y=1.Now, by using equation (77), we have(83)FN∗=Dx2+Dy2fx,y|x=y=1,FN∗=Dx2+Dy22x4y4+4x4y5+t−2x5y5+4x5y7+2t−2x5y8+2x7y8+2t−5x8y8|x=y=1,FN∗=32x4y4+32x4y4+64x4y5+100x4y5+25t−2x5y5+25t−2x5y5+100x5y7+196x5y7+50t−2x5y8+128t−2x5y8+98x7y8+128x7y8+642t−5x8y8+642t−5x8y8|x=y=1,FN∗=484t−346.(4)
M2nm is defined as(84)M2nm=SxSyNMG|x=y=1.Now, by using equation (77), we have(85)M2nm=SxSyfx,y|x=y=1,M2nm=SxSy2x4y4+4x4y5+t−2x5y5+4x5y7+2t−2x5y8+2x7y8+2t−5x8y8|x=y=1,M2nm=216x4y4+420x4y5+125t−2x5y5+435x5y7+240t−2x5y8+256x7y8+1642t−5x8y8|x=y=1,M2nm=1642t−5+9100t−2+1940.(5)
NRα is defined as(86)NRα=DxαDyαNMG|x=y=1.Now, by using equation (77), we have(87)NRα=DxαDyαfx,y|x=y=1,NRα=DxαDyα2x4y4+4x4y5+t−2x5y5+4x5y7+2t−2x5y8+2x7y8+2t−5x8y8|x=y=1,NRα=16α2x4y4+20α4x4y5+25αt−2x5y5+35α4x5y7+40α2t−2x5y8+56α2x7y8+64α2t−5x8y8|x=y=1,NRα=16α2+20α4+25αt−2+35α4+40α2t−2,+56α2+64α2t−5.(6)
ND3 is defined as(88)ND3=DxDyDx+DyNMG|x=y=1.Now, by using equation (77), we have(89)ND3=DxDyDx+Dyfx,y|x=y=1,ND3=DxDyDx+Dy2x4y4+4x4y5+t−2x5y5+4x5y7+2t−2x5y8+2x7y8+2t−5x8y8|x=y=1,ND3=DxDy8x4y4+8x4y4+16x4y5+20x4y5+5t−2x5y5+5t−2x5y5+20x5y7+28x5y7+10t−2x5y8+16t−2x5y8+14x7y8+16x7y8+82t−5x8y8+82t−5x8y8|x=y=1.ND3=256x4y4+720x4y5+250t−2x5y5+1680x5y7+1040t−2x5y8+1680x7y8+10242t−5x8y8|x=y=1,ND3=3338t−3364.(7)
ND5 is defined as(90)ND5=DxSy+SxDyNMG|x=y=1.Now, by using equation (77), we have(91)ND5=DxSy+SxDyfx,y|x=y=1,ND5=DxSy+SxDy2x4y4+4x4y5+t−2x5y5+4x5y7+2t−2x5y8+2x7y8+2t−5x8y8|x=y=1,ND5=2x4y4+2x4y4+165x4y5+204x4y5+t−2x5y5+t−2x5y5+207x5y7+285x5y7+108t−2x5y8+165t−2x5y8+148x7y8+167x7y8+2t−5x8y8+2t−5x8y8|x=y=1,ND5=22t−5+12920t−2+3457140.(8)
NH is defined as(92)NH=2SxJNMG|x=y=1.Now, by using equation (77), we have(93)NH=2SxJfx,y|x=y=1,NH=2SxJ2x4y4+4x4y5+t−2x5y5+4x5y7+2t−2x5y8+2x7y8+2t−5x8y8|x=y=1,NH=48x8+89x9+210t−2x10+812x12+413t−2x13+415x15+2162t−5x16|x=1,NH=182t−5+3365t−2+20990.(9)
NI is defined as(94)NI=SxJDxDyNMG|x=y=1.Now, by using equation (77), we have(95)NI=SxJDxDyfx,y|x=y=1,NI=SxJDxDy2x4y4+4x4y5+t−2x5y5+4x5y7+2t−2x5y8+2x7y8+2t−5x8y8|x=y=1,NI=SxJ32x4y4+80x4y5+25t−2x5y5+140x5y7+80t−2x5y8+112x7y8+642t−5x8y8|x=y=1,NI=328x8+809x9+2510t−2x10+14012x12+8013t−2x13+11215x15+64162t−5x16|x=1,NI=42t−5+22526t−2+144145.(10)
S is defined as(96)S=Sx3Q−2JDx3Dy3NMG|x=y=1.Now, by using equation (77), we have(97)S=Sx3Q−2JDx3Dy3fx,y|x=y=1,S=Sx3Q−2JDx3Dy32x4y4+4x4y5+t−2x5y5+4x5y7+2t−2x5y8+2x7y8+2t−5x8y8|x=y=1,S=8192216x6+32000343x7+15625512t−2x8+1715001000x10+1280001331t−2x11+3512322197x13+26214427442t−5x14|x=1,S=95.53352772t−5+126.6858726t−2+462.5892987.(11)
Fifth hyperM1 Zagreb index is defined as(98)HM1G5G=Dx2+Dy2+2DxDyNMG|x=y=1.Now, by using equation (77), we have(99)HM1G5Gt,s=Dx2+Dy2+2DxDyfx,y|x=y=1,HM1G5Gt,s=Dx2+Dy2+2DxDy2x4y4+4x4y5+t−2x5y5+4x5y7+2t−2x5y8+2x7y8+2t−5x8y8|x=y=1,HM1G5Gt,s=32x4y4+32x4y4+64x4y5+100x4y5+25t−2x5y5+25t−2x5y5+100x5y7+196x5y7+50t−2x5y8+128t−2x5y8+98x7y8+128x7y8+642t−5x8y8+642t−5x8y8+232x4y4+80x4y5+25t−2x5y5+140x5y7+80t−2x5y8+112x7y8+642t−5x8y8|x=y=1,HM1G5Gt,s=950t−678.(12)
Fifth hyperM2 Zagreb index is defined as(100)HM2G5G=DxDyDxDyNMG|x=y=1.Now, by using equation (77), we have(101)HM2G5Gt,s=DxDyDxDyfx,y|x=y=1,HM2G5Gt,s=DxDyDxDy2x4y4+4x4y5+t−2x5y5+4x5y7+2t−2x5y8+2x7y8+2t−5x8y8|x=y=1,HM2G5Gt,s=DxDy32x4y4+80x4y5+25t−2x5y5+140x5y7+80t−2x5y8+112x7y8+642t−5x8y8|x=y=1,HM2G5Gt,s=512x4y4+1600x4y5+625t−2x5y5+4900x5y7+3200t−2x5y8+6272x7y8+40962t−5x8y8|x=y=1,HM2G5Gt,s=12017t−14846.(13)
Fifth arithmetic-geometric index is defined as(102)AG5G=12Sx1/2Sy1/2Dx+DyNMG|x=y=1.Now, by using equation (77), we have(103)AG5Gt,s=12Sx1/2Sy1/2Dx+Dyfx,y|x=y=1,AG5Gt,s=12Sx1/2Sy1/2Dx+Dy2x4y4+4x4y5+t−2x5y5+4x5y7+2t−2x5y8+2x7y8+2t−5x8y8|x=y=1,AG5Gt,s=12Sx1/2Sy1/28x4y4+8x4y4+16x4y5+20x4y5+5t−2x5y5+5t−2x5y5+20x5y7+28x5y7+10t−2x5y8+16t−2x5y8+14x7y8+16x7y8+82t−5x8y8+82t−5x8y8|x=y=1,AG5Gt,s=16216x4y4+36220x4y5+10225t−2x5y5+48235x5y7+26240t−2x5y8+30256x7y8+162642t−5x8y8|x=y=1,AG5Gt,s=2t−5+3.055480479t−2+12.0861221.(14)
Fifth geometric-arithmetic index is defined as(104)GA5G=2SxJDx1/2DY1/2NMG|x=1.Now, by using equation (77), we have(105)GA5Gt,s=2SxJDx1/2Dy1/2fx,y|x=1,GA5Gt,s=2SxJDx1/2Dy1/22x4y4+4x4y5+t−2x5y5+4x5y7+2t−2x5y8+2x7y8+2t−5x8y8|x=1,GA5Gt,s=4168x8+8209x9+22510t−2x10+83512x12+44013t−2x13+45615x15+264162t−5x16|x=1,GA5Gt,s=2t−5+2.946017022t−2+11.91483576.


## 3.1. Graphene Networks

Graphene is an atomic scale honeycomb lattice made of the carbon atoms. Graphene is denoted byGt,s, where t is the number of rows of benzene rings and s is the number of benzene rings in each row. Graphene has 2st+2s+2t vertices and 3st+2s+2t−1 edges [33]. Figure 1 shows the graphene Gt,s.Figure 1
GrapheneGt,s.Theorem 1.
LetGt,sbe a graphene network, witht>1ands>1; then, its neighborhoodM-polynomial is given by(4)NMGt,s=4x4y5+tx5y5+8x5y7+2t−4x5y8+4s−8x6y7+2sx7y9+t−2x8y8+2t−4x8y9+3st−4s−4t+5x9y9.Proof.
Graphene has2st+2s+2t vertices and 3st+2s+2t−1 edges. The edge set of Gt,s, for t>1 and s>1, can be partitioned as follows:(5)E4,5=uv∈EGt,s:δu=4,δv=5=4=m4,5,E5,5=uv∈EGt,s:δu=5,δv=5=t=m5,5,E5,7=uv∈EGt,s:δu=5,δv=7=8=m5,7,E5,8=uv∈EGt,s:δu=5,δv=8=2t−4=m5,8,E6,7=uv∈EGt,s:δu=6,δv=7=4s−8=m6,7,E7,9=uv∈EGt,s:δu=7,δv=9=2s=m7,9,E8,8=uv∈EGt,s:δu=8,δv=8=t−2=m8,8,E8,9=uv∈EGt,s:δu=8,δv=9=2t−4=m8,9,E9,9=uv∈EGt,s:δu=9,δv=9=3st−4s−4t+5=m9,9.
Thus, the neighborhoodM -polynomial of Gt,s, for t>1 and s>1, is(6)NMGt,s=∑i≤jmi,jxiyj,NMGt,s=4x4y5+tx5y5+8x5y7+2t−4x5y8+4s−8x6y7+2sx7y9+t−2x8y8+2t−4x8y9+3st−4s−4t+5x9y9,
This is the required neighborhoodM -polynomial of Gt,s, for t>1 and s>1.Theorem 2.
LetGt,sbe a graphene network, witht=1ands>1; then, its neighborhoodM-polynomial is given by(7)NMGt,s=2x4y4+4x4y5+4x5y7+4s−8x6y7+s−1x7y7.Proof.
Graphene has2st+2s+2t vertices and 3st+2s+2t−1 edges. The edge set of Gt,s, for t=1 and s>1, can be partitioned as follows:(8)E4,4=uv∈EGt,s:δu=4,δv=4=2=m4,4,E4,5=uv∈EGt,s:δu=4,δv=5=4=m4,5,E5,7=uv∈EGt,s:δu=5,δv=7=4=m5,7,E6,7=uv∈EGt,s:δu=6,δv=7=4s−8=m6,7,E7,7=uv∈EGt,s:δu=7,δv=7=s−1=m7,7.
Thus, the neighborhoodM -polynomial of Gt,s, for t=1 and s>1, is(9)NMGt,s=∑i≤jmi,jxiyj,NMGt,s=2x4y4+4x4y5+4x5y7+4s−8x6y7+s−1x7y7.
This is the required neighborhoodM -polynomial of Gt,s, for t=1 and s>1.Theorem 3.
LetGt,sbe a graphene network, witht>2ands=1; then, its neighborhoodM-polynomial is given by(10)NMGt,s=2x4y4+4x4y5+t−2x5y5+4x5y7+2t−4x5y8+2x7y8+2t−5x8y8.Proof.
Graphene has2st+2s+2t vertices and 3st+2s+2t−1 edges. The edge set of Gt,s, for t>2 and s=1, can be partitioned as follows:(11)E4,4=uv∈EGt,s:δu=4,δv=4=2=m4,4,E4,5=uv∈EGt,s:δu=4,δv=5=4=m4,5,E5,5=uv∈EGt,s:δu=5,δv=5=t−2=m5,5,E5,7=uv∈EGt,s:δu=5,δv=7=4=m5,7,E5,8=uv∈EGt,s:δu=5,δv=8=2t−4=m5,8,E7,8=uv∈EGt,s:δu=7,δv=8=2=m7,8,E8,8=uv∈EGt,s:δu=8,δv=8=2t−5=m8,8.
Thus, the neighborhoodM -polynomial of Gt,s, for t>2 and s=1, is(12)NMGt,s=∑i≤jmi,jxiyj,NMGt,s=2x4y4+4x4y5+t−2x5y5+4x5y7+2t−4x5y8+2x7y8+2t−5x8y8,this is the required neighborhood M -polynomial of Gt,s, for t>2 and s=1.
Now, we calculate neighborhood degree-based graphical indices of graphene by using its neighborhoodM -polynomial.Corollary 1.
LetGt,sbe a graphene network, witht>1ands>1;then, its neighborhood degree-based graphical indices are given by(1)
M1′=54st+12s+14t−34(2)
M2∗=243st−30s−11t−147(3)
FN∗=486st−48s−2t−306(4)
M2nm=1/813st−4s−4t+5+2/21s−2+269/2880t−2+2/63s+1/25t+3/7(5)
NRα=20α4+25αt+35α8+40α2t−2+42α4s−2+63α2s+64αt−2+72α2t−2+81α3st−4s−4t+5(6)
ND3=4374st−1632s−1070t−2022(7)
ND5=23st−4s−4t+5+170/21s−2+943/90t−2+260/63s+2t+879/35(8)
NH=1/93st−4s−4t+5+8/13s−2+1181/1768t−2+1/4s+1/5t+20/9(9)
NI=9/23st−4s−4t+5+168/13s−2+4116/221t−2+63/8s+5/2t+290/9(10)
S=129.74633793st−4s−4t+5+222.6536439s−2+412.8858222t−2+729/4s+15625/512t+436.2944606(11)
HM1G5Gt,s=972st−108s−24t−600(12)
HM2G5Gt,s=19683st−11250s−7955t−5235(13)
AG5Gt,s=3st−4s−4t+5+4.011887099s−2+5.058949692t−2+2.015810523s+t+12.1384032(14)
GA5Gt,s=3st−4s−4t+5+842/13s−2+4.942553816t−2+37/4s+t+11.86333834Proof.
NM -Polynomial of Gt,s, for t>1 and s>1, is given by(13)NMGt,s=4x4y5+tx5y5+8x5y7+2t−2x5y8+4s−2x6y7+2sx7y9+t−2x8y8+2t−2x8y9+3st−4s−4t+5x9y9.
Let(14)NMGt,s=fx,y.
Then, we have(15)fx,y=NMGt,s=4x4y5+tx5y5+8x5y7+2t−2x5y8+4s−2x6y7+2sx7y9+t−2x8y8+2t−2x8y9+3st−4s−4t+5x9y9.(1)
M1′ is defined as(16)M1′=Dx+DyNMG|x=y=1.Now, by using equation (15), we have(17)M1′=Dx+Dyfx,y|x=y=1,M1′=Dx+Dy4x4y5+tx5y5+8x5y7+2t−2x5y8+4s−2x6y7+2sx7y9+t−2x8y8+2t−2x8y9+3st−4s−4t+5x9y9|x=y=1,M1′=16x4y5+20x4y5+5tx5y5+5tx5y5+40x5y7+56x5y7+10t−2x5y8+16t−2x5y8+24s−2x6y7+28s−2x6y7+14sx7y9+18sx7y9+8t−2x8y8+8t−2x8y8+16t−2x8y9+18t−2x8y9+93st−4s−4t+5x9y9+93st−4s−4t+5x9y9|x=y=1,M1′=54st+12s+14t−34.(2)
M2∗ is defined as(18)M2∗=DxDyNMG|x=y=1.Now, by using equation (15), we have(19)M2∗=DxDyfx,y|x=y=1,M2∗=DxDy4x4y5+tx5y5+8x5y7+2t−2x5y8+4s−2x6y7+2sx7y9+t−2x8y8+2t−2x8y9+3st−4s−4t+5x9y9|x=y=1,M2∗=80x4y5+25tx5y5+280x5y7+80t−2x5y8+168s−2x6y7+126sx7y9+64t−2x8y8+144t−2x8y9+813st−4s−4t+5x9y9|x=y=1,M2∗=243st−30s−11t−147.(3)
FN∗ is defined as(20)FN∗=Dx2+Dy2NMG|x=y=1.Now, by using equation (15), we have(21)FN∗=Dx2+Dy2fx,y|x=y=1,FN∗=Dx2+Dy24x4y5+tx5y5+8x5y7+2t−2x5y8+4s−2x6y7+2sx7y9+t−2x8y8+2t−2x8y9+3st−4s−4t+5x9y9|x=y=1,FN∗=64x4y5+100x4y5+25tx5y5+25tx5y5+200x5y7+392x5y7+50t−2x5y8+128t−2x5y8+144s−2x6y7+196s−2x6y7+98sx7y9+162sx7y9+64t−2x8y8+64t−2x8y8+128t−2x8y9+162t−2x8y9+813st−4s−4t+5x9y9+813st−4s−4t+5x9y9|x=y=1,FN∗=486st−48s−2t−306.(4)
M2nm is defined as(22)M2nm=SxSyNMG|x=y=1.Now, by using equation (15), we have(23)M2nm=SxSyfx,y|x=y=1,M2nm=SxSy4x4y5+tx5y5+8x5y7+2t−2x5y8+4s−2x6y7+2sx7y9+t−2x8y8+2t−2x8y9+3st−4s−4t+5x9y9|x=y=1,M2nm=420x4y5+125tx5y5+835x5y7+240t−2x5y8+442s−2x6y7+263sx7y9+164t−2x8y8+272t−2x8y9+1813st−4s−4t+5x9y9|x=y=1,M2nm=1813st−4s−4t+5+221s−2+2692880t−2+263s+125t+37.(5)
NRα is defined as(24)NRα=DxαDyαNMG|x=y=1.Now, by using equation (15), we have(25)NRα=DxαDyαfx,y|x=y=1,NRα=DxαDyα4x4y5+tx5y5+8x5y7+2t−2x5y8+4s−2x6y7+2sx7y9+t−2x8y8+2t−2x8y9+3st−4s−4t+5x9y9|x=y=1,NRα=20α4x4y5+25αtx5y5+35α8x5y7+40α2t−2x5y8+42α4s−2x6y7+63α2sx7y9+64αt−2x8y8+72α2t−2x8y9+81α3st−4s−4t+5x9y9|x=y=1,NRα=20α4+25αt+35α8+40α2t−2+42α4s−2+63α2s+64αt−2+72α2t−2+81α3st−4s−4t+5.(6)
ND3 is defined as(26)ND3=DxDyDx+DyNMG|x=y=1.Now, by using equation (15), we have(27)ND3=DxDyDx+Dyfx,y|x=y=1,ND3=DxDyDx+Dy4x4y5+tx5y5+8x5y7+2t−2x5y8+4s−2x6y7+2sx7y9+t−2x8y8+2t−2x8y9+3st−4s−4t+5x9y9|x=y=1,ND3=DxDy16x4y5+20x4y5+5tx5y5+5tx5y5+40x5y7+56x5y7+10t−2x5y8+16t−2x5y8+24s−2x6y7+28s−2x6y7+14sx7y9+18sx7y9+8t−2x8y8+8t−2x8y8+16t−2x8y9+18t−2x8y9+93st−4s−4t+5x9y9+93st−4s−4t+5x9y9|x=y=1,ND3=720x4y5+250tx5y5+3360x5y7+1040t−2x5y8+2184s−2x6y7+2016sx7y9+1024t−2x8y8+2448t−2x8y9+14583st−4s−4t+5x9y9|x=y=1,ND3=4374st−1632s−1070t−2022.(7)
ND5 is defined as(28)ND5=DxSy+SxDyNMG|x=y=1.Now, by using equation (15), we have(29)ND5=DxSy+SxDyfx,y|x=y=1.ND5=DxSy+SxDy4x4y5+tx5y5+8x5y7+2t−2x5y8+4s−2x6y7+2sx7y9+t−2x8y8+2t−2x8y9+3st−4s−4t+5x9y9|x=y=1,ND5=165x4y5+204x4y5+tx5y5+tx5y5+407x5y7+565x5y7+108t−2x5y8+165t−2x5y8+247s−2x6y7+286s−2x6y7+149sx7y9+187sx7y9+t−2x8y8+t−2x8y8+169t−2x8y9+188t−2x8y9+3st−4s−4t+5x9y9+3st−4s−4t+5x9y9|x=y=1,ND5=23st−4s−4t+5+17021s−2+94390t−2+26063s+2t+87935.(8)
NH is defined as(30)NH=2SxJNMG|x=y=1.Now, by using equation (15), we have(31)NH=2SxJfx,y|x=y=1,NH=2SxJ4x4y5+tx5y5+8x5y7+2t−2x5y8+4s−2x6y7+2sx7y9+t−2x8y8+2t−2x8y9+3st−4s−4t+5x9y9|x=y=1,NH=89x9+210tx10+1612x12+413t−2x13+813s−2x13+416sx16+216t−2x16+417t−2x17+2183st−4s−4t+5x18|x=1,NH=193st−4s−4t+5+813s−2+11811768t−2+14s+15t+209.(9)
NI is defined as(32)NI=SxJDxDyNMG|x=y=1.Now, by using equation (15), we have(33)NI=SxJDxDyfx,y|x=y=1,NI=SxJDxDy4x4y5+tx5y5+8x5y7+2t−2x5y8+4s−2x6y7+2sx7y9+t−2x8y8+2t−2x8y9+3st−4s−4t+5x9y9|x=y=1,NI=SxJ80x4y5+25tx5y5+280x5y7+80t−2x5y8+168s−2x6y7+126sx7y9+64t−2x8y8+144t−2x8y9+813st−4s−4t+5x9y9|x=y=1,NI=809x9+2510tx10+28012x12+8013t−2x13+16813s−2x13+12616sx16+6416t−2x16+14417t−2x17+81183st−4s−4t+5x18|x=1,NI=923st−4s−4t+5+16813s−2+4116221t−2+638s+52t+2909.(10)
S is defined as(34)S=Sx3Q−2JDx3Dy3NMG|x=y=1.Now, by using equation (15), we have(35)S=Sx3Q−2JDx3Dy3fx,y|x=y=1,S=Sx3Q−2JDx3Dy34x4y5+tx5y5+8x5y7+2t−2x5y8+4s−2x6y7+2sx7y9+t−2x8y8+2t−2x8y9+3st−4s−4t+5x9y9|x=y=1,S=32000343x7+15625512tx8+3430001000x10+1280001331t−2x11+2963521331s−2x11+5000942744sx14+2621442744t−2x14+7464963375t−2x15+53144140963st−4s−4t+5x16|x=1,S=129.74633793st−4s−4t+5+222.6536439s−2+412.8858222t−2+7294s+15625512t+436.2944606.(11)
Fifth hyperM1 Zagreb index is defined as(36)HM1G5G=Dx2+Dy2+2DxDyNMG|x=y=1.Now, by using equation (15), we have(37)HM1G5Gt,s=Dx2+Dy2+2DxDyfx,y|x=y=1,HM1G5Gt,s=Dx2+Dy2+2DxDy4x4y5+tx5y5+8x5y7+2t−2x5y8+4s−2x6y7+2sx7y9+t−2x8y8+2t−2x8y9+3st−4s−4t+5x9y9|x=y=1,HM1G5Gt,s=64x4y5+100x4y5+25tx5y5+25tx5y5+200x5y7+392x5y7+50t−2x5y8+128t−2x5y8+144s−2x6y7+196s−2x6y7+98sx7y9+162sx7y9+64t−2x8y8+64t−2x8y8+128t−2x8y9+162t−2x8y9+813st−4s−4t+5x9y9+813st−4s−4t+5x9y9+280x4y5+25tx5y5+280x5y7+80t−2x5y8+168s−2x6y7+126sx7y9+64t−2x8y8+144t−2x8y9+813st−4s−4t+5x9y9|x=y=1,HM1G5Gt,s=972st−108s−24t−600.(12)
Fifth hyperM2 Zagreb index is defined as(38)HM2G5G=DxDyDxDyNMG|x=y=1.Now, by using equation (15), we have(39)HM2G5Gt,s=DxDyDxDyfx,y|x=y=1,HM2G5Gt,s=DxDyDxDy4x4y5+tx5y5+8x5y7+2t−2x5y8+4s−2x6y7+2sx7y9+t−2x8y8+2t−2x8y9+3st−4s−4t+5x9y9|x=y=1,HM2G5Gt,s=DxDy80x4y5+25tx5y5+280x5y7+80t−2x5y8+168s−2x6y7+126sx7y9+64t−2x8y8+144t−2x8y9+813st−4s−4t+5x9y9|x=y=1,HM2G5Gt,s=1600x4y5+625tx5y5+9800x5y7+3200t−2x5y8+7056s−2x6y7+7938sx7y9+4096t−2x8y8+10368t−2x8y9+65613st−4s−4t+5x9y9|x=y=1,HM2G5Gt,s=19683st−11250s−7955t−5235.(13)
Fifth arithmetic-geometric index is defined as(40)AG5G=12Sx1/2Sy1/2Dx+DyNMG|x=y=1.Now, by using equation (15), we have(41)AG5Gt,s=12Sx1/2Sy1/2Dx+Dyfx,y|x=y=1,AG5Gt,s=12Sx1/2Sy1/2Dx+Dy4x4y5+tx5y5+8x5y7+2t−2x5y8+4s−2x6y7+2sx7y9+t−2x8y8+2t−2x8y9+3st−4s−4t+5x9y9|x=y=1,AG5Gt,s=12Sx1/2Sy1/216x4y5+20x4y5+5tx5y5+5tx5y5+40x5y7+56x5y7+10t−2x5y8+16t−2x5y8+24s−2x6y7+28s−2x6y7+14sx7y9+18sx7y9+8t−2x8y8+8t−2x8y8+16t−2x8y9+18t−2x8y9+93st−4s−4t+5x9y9+93st−4s−4t+5x9y9|x=y=1,AG5Gt,s=36220x4y5+10225tx5y5+96235x5y7+26240t−2x5y8+52242s−2x6y7+32263sx7y9+16264t−2x8y8+34272t−2x8y9+182813st−4s−4t+5x9y9|x=y=1,AG5Gt,s=3st−4s−4t+5+4.011887099s−2+5.058949692t−2+2.015810523s+t+12.1384032.(14)
Fifth geometric-arithmetic index is defined as(42)GA5G=2SxJDx1/2Dy1/2NMG|x=1,Now, by using equation (15), we have(43)GA5Gt,s=2SxJDx1/2Dy1/2fx,y|x=1,GA5Gt,s=2SxJDx1/2Dy1/24x4y5+tx5y5+8x5y7+2t−2x5y8+4s−2x6y7+2sx7y9+t−2x8y8+2t−2x8y9+3st−4s−4t+5x9y9|x=1,GA5Gt,s=2SxJ420x4y5+25tx5y5+835x5y7+240t−2x5y8+442s−2x6y7+263sx7y9+64t−2x8y8+272t−2x8y9+813st−4s−4t+5x9y9|x=1,GA5Gt,s=8209x9+22510tx10+163512x12+44013t−2x13c+84213s−2x13+46316sx16+26416t−2x16+47217t−2x17+281183st−4s−4t+5x18|x=1,GA5Gt,s=3st−4s−4t+5+84213s−2+4.942553816t−2+374s+t+11.86333834.Corollary 2.
LetGt,sbe a graphene network, witht=1ands>1; then, its neighborhood degree-based graphical indices are given by(1)
M1′=66s−18(2)
M2∗=217s−133(3)
FN∗=438s−254(4)
M2nm=2/21s−2+1/49s−1+123/280(5)
NRα=16α2+20α4+35α4+42α4s−2+49αs−1(6)
ND3=2870s−2398(7)
ND5=170/21s−2+2s−1+723/35(8)
NH=8/13s−2+1/7s−1+37/18(9)
NI=168/13s−2+7/2s−1+221/9(10)
S=222.6536439s−2+68.08391204s−1+302.7203866(11)
HM1G5Gt,s=872s−520(12)
HM2G5Gt,s=9457s−9501(13)
AG5Gt,s=1342/21s−2+s−1+10.08166278(14)
GA5Gt,s=842/13s−2+s−1+9.919285149Proof.
NM -Polynomial of Gt,s, for t=1 and s>1, is given by(44)NMGt,s=2x4y4+4x4y5+4x5y7+4s−2x6y7+s−1x7y7.
Let,(45)NMGt,s=fx,y,then, we have(46)fx,y=NMGt,s=2x4y4+4x4y5+4x5y7+4s−2x6y7+s−1x7y7.(1)
M1′ is defined as(47)M1′=Dx+DyNMG|x=y=1.Now, by using equation (46), we have(48)M1′=Dx+Dyfx,y|x=y=1,M1′=Dx+Dy2x4y4+4x4y5+4x5y7+4s−2x6y7+s−1x7y7|x=y=1,M1′=8x4y4+8x4y4+16x4y5+20x4y5+20x5y7+28x5y7+24s−2x6y7+28s−2x6y7+7s−1x7y7+7s−1x7y7|x=y=1,M1′=66s−18.(2)
M2∗ is defined as(49)M2∗=DxDyNMG|x=y=1.Now, by using equation (46), we have(50)M2∗=DxDyfx,y|x=y=1,M2∗=DxDy2x4y4+4x4y5+4x5y7+4s−2x6y7+s−1x7y7|x=y=1,M2∗=32x4y4+80x4y5+140x5y7+168s−2x6y7+49s−1x7y7|x=y=1,M2∗=217s−133.(3)
FN∗ is defined as(51)FN∗=Dx2+Dy2NMG|x=y=1.Now, by using equation (46), we have(52)FN∗=Dx2+Dy2fx,y|x=y=1,FN∗=Dx2+Dy22x4y4+4x4y5+4x5y7+4s−2x6y7+s−1x7y7|x=y=1,FN∗=32x4y4+32x4y4+64x4y5+100x4y5+100x5y7+196x5y7+144s−2x6y7+196s−2x6y7+490s−1x7y7+49s−1x7y7|x=y=1,FN∗=438s−254.(4)
M2nm is defined as(53)M2nm=SxSyNMG|x=y=1.Now, by using equation (46), we have(54)M2nm=SxSyfx,y|x=y=1,M2nm=SxSy2x4y4+4x4y5+4x5y7+4s−2x6y7+s−1x7y7|x=y=1,M2nm=216x4y4+420x4y5+435x5y7+442s−2x6y7+149s−1x7y7|x=y=1,M2nm=221s−2+149s−1+123280.(5)
NRα is defined as(55)NRα=DxαDyαNMG|x=y=1.Now, by using equation (46), we have(56)NRα=DxαDyαfx,y|x=y=1,NRα=DxαDyα2x4y4+4x4y5+4x5y7+4s−2x6y7+s−1x7y7|x=y=1,NRα=16α2x4y4+20α4x4y5+35α4x5y7+42α4s−2x6y7+49αs−1x7y7|x=y=1,NRα=16α2+20α4+35α4+42α4s−2+49αs−1.(6)
ND3 is defined as(57)ND3=DxDyDx+DyNMG|x=y=1.Now, by using equation (46), we have(58)ND3=DxDyDx+Dyfx,y|x=y=1,ND3=DxDyDx+Dy2x4y4+4x4y5+4x5y7+4s−2x6y7+s−1x7y7|x=y=1,ND3=DxDy8x4y4+8x4y4+16x4y5+20x4y5+20x5y7+28x5y7+24s−2x6y7+28s−2x6y7+7s−1x7y7+7s−1x7y7|x=y=1,ND3=256x4y4+720x4y5+1680x5y7+2184s−2x6y7+686s−1x7y7|x=y=1,ND3=2870s−2398.(7)
ND5 is defined as(59)ND5=DxSy+SxDyNMG|x=y=1.Now, by using equation (46), we have(60)ND5=DxSy+SxDyfx,y|x=y=1,ND5=DxSy+SxDy2x4y4+4x4y5+4x5y7+4s−2x6y7+s−1x7y7|x=y=1,ND5=2x4y4+2x4y4+165x4y5+204x4y5+207x5y7+285x5y7+247s−2x6y7+286s−2x6y7+s−1x7y7+s−1x7y7|x=y=1,ND5=17021s−2+2s−1+72335.(8)
NH is defined as(61)NH=2SxJNMG|x=y=1.Now, by using equation (46), we have(62)NH=2SxJfx,y|x=y=1,NH=2SxJ2x4y4+4x4y5+4x5y7+4s−2x6y7+s−1x7y7|x=y=1,NH=48x8+89x9+812x12+813s−2x13+214s−1x14|x=1,NH=813s−2+17s−1+3718.(9)
NI is defined as(63)NI=SxJDxDyNMG|x=y=1.Now, by using equation (46), we have(64)NI=SxJDxDyfx,y|x=y=1,NI=SxJDxDy2x4y4+4x4y5+4x5y7+4s−2x6y7+s−1x7y7|x=y=1,NI=SxJ32x4y4+80x4y5+140x5y7+168s−2x6y7+49s−1x7y7|x=y=1,NI=328x8+809x9+14012x12+16813s−2x13+4914s−1x14|x=1,NI=16813s−2+72s−1+2219.(10)
S is defined as(65)S=Sx3Q−2JDx3Dy3NMG|x=y=1.Now, by using equation (46), we have(66)S=Sx3Q−2JDx3Dy3fx,y|x=y=1,S=Sx3Q−2JDx3Dy32x4y4+4x4y5+4x5y7+4s−2x6y7+s−1x7y7|x=y=1,S=8192216x6+32000343x7+1715001000x10+2963521331s−2x11+1176491728s−1x12|x=1,S=222.6536439s−2+68.08391204s−1+302.7203866.(11)
Fifth hyperM1 Zagreb index is defined as(67)HM1G5G=Dx2+Dy2+2DxDyNMG|x=y=1.Now, by using equation (46), we have(68)HM1G5Gt,s=Dx2+Dy2+2DxDyfx,y|x=y=1,HM1G5Gt,s=Dx2+Dy2+2DxDy2x4y4+4x4y5+4x5y7+4s−2x6y7+s−1x7y7|x=y=1,HM1G5Gt,s=32x4y4+32x4y4+64x4y5+100x4y5+100x5y7+196x5y7+144s−2x6y7+196s−2x6y7+49s−1x7y7+49s−1x7y7+232x4y4+80x4y5+140x5y7+168s−2x6y7+49s−1x7y7|x=y=1,HM1G5Gt,s=872s−520.(12)
Fifth hyperM2 Zagreb index is defined as(69)HM2G5G=DxDyDxDyNMG|x=y=1.Now, by using equation (46), we have(70)HM2G5Gt,s=DxDyDxDyfx,y|x=y=1,HM2G5Gt,s=DxDyDxDy2x4y4+4x4y5+4x5y7+4s−2x6y7+s−1x7y7|x=y=1,HM2G5Gt,s=DxDy32x4y4+80x4y5+140x5y7+168s−2x6y7+49s−1x7y7|x=y=1,HM2G5Gt,s=512x4y4+1600x4y5+4900x5y7+7056s−2x6y7+2401s−1x7y7|x=y=1,HM2G5Gt,s=9457s−9501.(13)
Fifth arithmetic-geometric index is defined as(71)AG5G=12Sx1/2Sy1/2Dx+DyNMG|x=y=1.Now, by using equation (46), we have(72)AG5Gt,s=12Sx1/2Sy1/2Dx+Dyfx,y|x=y=1,AG5Gt,s=12Sx1/2Sy1/2Dx+Dy2x4y4+4x4y5+4x5y7+4s−2x6y7+s−1x7y7|x=y=1,AG5Gt,s=12Sx1/2Sy1/28x4y4+8x4y4+16x4y5+20x4y5+20x5y7+28x5y7+24s−2x6y7+28s−2x6y7+7s−1x7y7+7s−1x7y7|x=y=1,AG5Gt,s=16216x4y4+36220x4y5+48235x5y7+52242s−2x6y7+14249s−1x7y7|x=y=1,AG5Gt,s=134221s−2+s−1+10.08166278.(14)
Fifth geometric-arithmetic index is defined as(73)GA5G=2SxJDx1/2Dy1/2NMG|x=1.Now, by using equation (46), we have(74)GA5Gt,s=2SxJDx1/2Dy1/2fx,y|x=1,GA5Gt,s=2SxJDx1/2Dy1/22x4y4+4x4y5+4x5y7+4s−2x6y7+s−1x7y7|x=1,GA5Gt,s=4168x8+8209x9+83512x12+84213s−2x13+24914s−1x14|x=1,GA5Gt,s=84213s−2+s−1+9.919285149.Corollary 3.
LetGt,sbe a graphene network, witht>2ands=1; then, its neighborhood degree-based graphical indices are given by(1)
M1′=68t−22(2)
M2∗=233t−166(3)
FN∗=484t−346(4)
M2nm=1/642t−5+9/100t−2+19/40(5)
NRα=16α2+20α4+25αt−2+35α4+40α2t−2+56α2+64α2t−5(6)
ND3=3338t−3364(7)
ND5=22t−5+129/20t−2+3457/140(8)
NH=1/82t−5+33/65t−2+209/90(9)
NI=42t−5+225/26t−2+1441/45(10)
S=95.53352772t−5+126.6858726t−2+462.5892987(11)
HM1G5Gt,s=950t−678(12)
HM2G5Gt,s=12017t−14846(13)
AG5Gt,s=2t−5+3.055480479t−2+12.0861221(14)
GA5Gt,s=2t−5+2.946017022t−2+11.91483576Proof.
NM -Polynomial of Gt,s, for t>2 and s=1, is given by(75)NMGt,s=2x4y4+4x4y5+t−2x5y5+4x5y7+2t−2x5y8+2x7y8+2t−5x8y8.
Let,(76)NMGt,s=fx,y.
Then, we have(77)fx,y=NMGt,s=2x4y4+4x4y5+t−2x5y5+4x5y7+2t−2x5y8+2x7y8+2t−5x8y8.(1)
M1′ is defined as(78)M1′=Dx+DyNMG|x=y=1.Now, by using equation (77), we have(79)M1′=Dx+Dyfx,y|x=y=1,M1′=Dx+Dy2x4y4+4x4y5+t−2x5y5+4x5y7+2t−2x5y8+2x7y8+2t−5x8y8|x=y=1,M1′=8x4y4+8x4y4+16x4y5+20x4y5+5t−2x5y5+5t−2x5y5+20x5y7+28x5y7+10t−2x5y8+16t−2x5y8+14x7y8+16x7y8+82t−5x8y8+82t−5x8y8|x=y=1,M1′=68t−22.(2)
M2∗ is defined as(80)M2∗=DxDyNMG|x=y=1.Now, by using equation (77), we have(81)M2∗=DxDyfx,y|x=y=1,M2∗=DxDy2x4y4+4x4y5+t−2x5y5+4x5y7+2t−2x5y8+2x7y8+2t−5x8y8|x=y=1,M2∗=32x4y4+80x4y5+25t−2x5y5+140x5y7+80t−2x5y8+112x7y8+642t−5x8y8|x=y=1,M2∗=233t−166.(3)
FN∗ is defined as(82)FN∗=Dx2+Dy2NMG|x=y=1.Now, by using equation (77), we have(83)FN∗=Dx2+Dy2fx,y|x=y=1,FN∗=Dx2+Dy22x4y4+4x4y5+t−2x5y5+4x5y7+2t−2x5y8+2x7y8+2t−5x8y8|x=y=1,FN∗=32x4y4+32x4y4+64x4y5+100x4y5+25t−2x5y5+25t−2x5y5+100x5y7+196x5y7+50t−2x5y8+128t−2x5y8+98x7y8+128x7y8+642t−5x8y8+642t−5x8y8|x=y=1,FN∗=484t−346.(4)
M2nm is defined as(84)M2nm=SxSyNMG|x=y=1.Now, by using equation (77), we have(85)M2nm=SxSyfx,y|x=y=1,M2nm=SxSy2x4y4+4x4y5+t−2x5y5+4x5y7+2t−2x5y8+2x7y8+2t−5x8y8|x=y=1,M2nm=216x4y4+420x4y5+125t−2x5y5+435x5y7+240t−2x5y8+256x7y8+1642t−5x8y8|x=y=1,M2nm=1642t−5+9100t−2+1940.(5)
NRα is defined as(86)NRα=DxαDyαNMG|x=y=1.Now, by using equation (77), we have(87)NRα=DxαDyαfx,y|x=y=1,NRα=DxαDyα2x4y4+4x4y5+t−2x5y5+4x5y7+2t−2x5y8+2x7y8+2t−5x8y8|x=y=1,NRα=16α2x4y4+20α4x4y5+25αt−2x5y5+35α4x5y7+40α2t−2x5y8+56α2x7y8+64α2t−5x8y8|x=y=1,NRα=16α2+20α4+25αt−2+35α4+40α2t−2,+56α2+64α2t−5.(6)
ND3 is defined as(88)ND3=DxDyDx+DyNMG|x=y=1.Now, by using equation (77), we have(89)ND3=DxDyDx+Dyfx,y|x=y=1,ND3=DxDyDx+Dy2x4y4+4x4y5+t−2x5y5+4x5y7+2t−2x5y8+2x7y8+2t−5x8y8|x=y=1,ND3=DxDy8x4y4+8x4y4+16x4y5+20x4y5+5t−2x5y5+5t−2x5y5+20x5y7+28x5y7+10t−2x5y8+16t−2x5y8+14x7y8+16x7y8+82t−5x8y8+82t−5x8y8|x=y=1.ND3=256x4y4+720x4y5+250t−2x5y5+1680x5y7+1040t−2x5y8+1680x7y8+10242t−5x8y8|x=y=1,ND3=3338t−3364.(7)
ND5 is defined as(90)ND5=DxSy+SxDyNMG|x=y=1.Now, by using equation (77), we have(91)ND5=DxSy+SxDyfx,y|x=y=1,ND5=DxSy+SxDy2x4y4+4x4y5+t−2x5y5+4x5y7+2t−2x5y8+2x7y8+2t−5x8y8|x=y=1,ND5=2x4y4+2x4y4+165x4y5+204x4y5+t−2x5y5+t−2x5y5+207x5y7+285x5y7+108t−2x5y8+165t−2x5y8+148x7y8+167x7y8+2t−5x8y8+2t−5x8y8|x=y=1,ND5=22t−5+12920t−2+3457140.(8)
NH is defined as(92)NH=2SxJNMG|x=y=1.Now, by using equation (77), we have(93)NH=2SxJfx,y|x=y=1,NH=2SxJ2x4y4+4x4y5+t−2x5y5+4x5y7+2t−2x5y8+2x7y8+2t−5x8y8|x=y=1,NH=48x8+89x9+210t−2x10+812x12+413t−2x13+415x15+2162t−5x16|x=1,NH=182t−5+3365t−2+20990.(9)
NI is defined as(94)NI=SxJDxDyNMG|x=y=1.Now, by using equation (77), we have(95)NI=SxJDxDyfx,y|x=y=1,NI=SxJDxDy2x4y4+4x4y5+t−2x5y5+4x5y7+2t−2x5y8+2x7y8+2t−5x8y8|x=y=1,NI=SxJ32x4y4+80x4y5+25t−2x5y5+140x5y7+80t−2x5y8+112x7y8+642t−5x8y8|x=y=1,NI=328x8+809x9+2510t−2x10+14012x12+8013t−2x13+11215x15+64162t−5x16|x=1,NI=42t−5+22526t−2+144145.(10)
S is defined as(96)S=Sx3Q−2JDx3Dy3NMG|x=y=1.Now, by using equation (77), we have(97)S=Sx3Q−2JDx3Dy3fx,y|x=y=1,S=Sx3Q−2JDx3Dy32x4y4+4x4y5+t−2x5y5+4x5y7+2t−2x5y8+2x7y8+2t−5x8y8|x=y=1,S=8192216x6+32000343x7+15625512t−2x8+1715001000x10+1280001331t−2x11+3512322197x13+26214427442t−5x14|x=1,S=95.53352772t−5+126.6858726t−2+462.5892987.(11)
Fifth hyperM1 Zagreb index is defined as(98)HM1G5G=Dx2+Dy2+2DxDyNMG|x=y=1.Now, by using equation (77), we have(99)HM1G5Gt,s=Dx2+Dy2+2DxDyfx,y|x=y=1,HM1G5Gt,s=Dx2+Dy2+2DxDy2x4y4+4x4y5+t−2x5y5+4x5y7+2t−2x5y8+2x7y8+2t−5x8y8|x=y=1,HM1G5Gt,s=32x4y4+32x4y4+64x4y5+100x4y5+25t−2x5y5+25t−2x5y5+100x5y7+196x5y7+50t−2x5y8+128t−2x5y8+98x7y8+128x7y8+642t−5x8y8+642t−5x8y8+232x4y4+80x4y5+25t−2x5y5+140x5y7+80t−2x5y8+112x7y8+642t−5x8y8|x=y=1,HM1G5Gt,s=950t−678.(12)
Fifth hyperM2 Zagreb index is defined as(100)HM2G5G=DxDyDxDyNMG|x=y=1.Now, by using equation (77), we have(101)HM2G5Gt,s=DxDyDxDyfx,y|x=y=1,HM2G5Gt,s=DxDyDxDy2x4y4+4x4y5+t−2x5y5+4x5y7+2t−2x5y8+2x7y8+2t−5x8y8|x=y=1,HM2G5Gt,s=DxDy32x4y4+80x4y5+25t−2x5y5+140x5y7+80t−2x5y8+112x7y8+642t−5x8y8|x=y=1,HM2G5Gt,s=512x4y4+1600x4y5+625t−2x5y5+4900x5y7+3200t−2x5y8+6272x7y8+40962t−5x8y8|x=y=1,HM2G5Gt,s=12017t−14846.(13)
Fifth arithmetic-geometric index is defined as(102)AG5G=12Sx1/2Sy1/2Dx+DyNMG|x=y=1.Now, by using equation (77), we have(103)AG5Gt,s=12Sx1/2Sy1/2Dx+Dyfx,y|x=y=1,AG5Gt,s=12Sx1/2Sy1/2Dx+Dy2x4y4+4x4y5+t−2x5y5+4x5y7+2t−2x5y8+2x7y8+2t−5x8y8|x=y=1,AG5Gt,s=12Sx1/2Sy1/28x4y4+8x4y4+16x4y5+20x4y5+5t−2x5y5+5t−2x5y5+20x5y7+28x5y7+10t−2x5y8+16t−2x5y8+14x7y8+16x7y8+82t−5x8y8+82t−5x8y8|x=y=1,AG5Gt,s=16216x4y4+36220x4y5+10225t−2x5y5+48235x5y7+26240t−2x5y8+30256x7y8+162642t−5x8y8|x=y=1,AG5Gt,s=2t−5+3.055480479t−2+12.0861221.(14)
Fifth geometric-arithmetic index is defined as(104)GA5G=2SxJDx1/2DY1/2NMG|x=1.Now, by using equation (77), we have(105)GA5Gt,s=2SxJDx1/2Dy1/2fx,y|x=1,GA5Gt,s=2SxJDx1/2Dy1/22x4y4+4x4y5+t−2x5y5+4x5y7+2t−2x5y8+2x7y8+2t−5x8y8|x=1,GA5Gt,s=4168x8+8209x9+22510t−2x10+83512x12+44013t−2x13+45615x15+264162t−5x16|x=1,GA5Gt,s=2t−5+2.946017022t−2+11.91483576.


## 4. NeighborhoodM -Polynomial of the Line Graph of the Subdivision Graph of Graphene Networks

In this section, we find neighborhoodM -polynomial of the line graph of the subdivision graph of graphene and calculate neighborhood degree-based graphical indices of the line graph of the subdivision graph of graphene by using its neighborhood M -polynomial.


### 4.1. Line Graph of the Subdivision Graph of Graphene Networks

The subdivision graph of grapheneGt,s is denoted by SGt,s, and the line graph of the subdivision graph of graphene is denoted by LSGt,s. The line graph of the subdivision graph of graphene has 23st+2s+2t−1 vertices and 9st+4s+4t−5 edges [34]. The subdivision graph of Gt,s and the line graph of the subdivision graph of Gt,s are shown in Figures 2 and 3 respectively.Figure 2
Subdivision graph ofGt,s.Figure 3
Line graph of the subdivision graph ofGt,s.Theorem 4.
LetLSGt,sbe the line graph of the subdivision graph of grapheneGt,s, witht≠1ands>1; then, its neighborhoodM-polynomial is given by(106)NMLSGt,s=t+6x4y4+2t+2x4y5+2s−2x5y5+22s+t−2x5y8+2sx8y8+4s+t−2x8y9+9st−8s−5t+1x9y9.Proof.
The line graph of the subdivision graph of graphene has23st+2s+2t−1 vertices and 9st+4s+4t−5 edges. The edge set of LSGt,s, for t≠1 and s>1, can be partitioned as follows:(107)E4,4=uv∈ELSGt,s:δu=4,δv=4=t+6=m4,4,E4,5=uv∈ELSGt,s:δu=4,δv=5=2t+2=m4,5,E5,5=uv∈ELSGt,s:δu=5,δv=5=2s−2=m5,5,E5,8=uv∈ELSGt,s:δu=5,δv=8=22s+t−2=m5,8,E8,8=uv∈ELSGt,s:δu=8,δv=8=2s=m8,8.E8,9=uv∈ELSGt,s:δu=8,δv=9=4s+t−2=m8,9,E9,9=uv∈ELSGt,s:δu=9,δv=9=9st−8s−5t+1=m9,9,
Thus, the neighborhoodM -polynomial of LSGt,s, for t≠1 and s>1, is(108)NMLSGt,s=∑i≤jmi,jxiyjm,NMLSGt,s=t+6x4y4+2t+2x4y5+2s−2x5y5+22s+t−2x5y8+2sx8y8+4s+t−2x8y9+9st−8s−5t+1x9y9.
This is the required neighborhoodM -polynomial of LSGt,s, for t≠1 and s>1.Theorem 5.
LetLSGt,sbe the line graph of the subdivision graph of grapheneGt,s, witht≠1ands=1; then, its neighborhoodM-polynomial is given by(109)NMLSGt,s=t+8x4y4+2tx4y5+2tx5y8+2x8y8+4t−1x8y9+4t−7x9y9.Proof.
The line graph of the subdivision graph of graphene has23st+2s+2t−1 vertices and 9st+4s+4t−5 edges. The edge set of LSGt,s, for t≠1 and s=1, can be partitioned as follows:'(110)E4,4=uv∈ELSGt,s:δu=4,δv=4=t+8=m4,4,E4,5=uv∈ELSGt,s:δu=4,δv=5=2t=m4,5,E5,8=uv∈ELSGt,s:δu=5,δv=8=2t=m5,8,E8,8=uv∈ELSGt,s:δu=8,δv=8=2=m8,8,E8,9=uv∈ELSGt,s:δu=8,δv=9=4t−1=m8,9,E9,9=uv∈ELSGt,s:δu=9,δv=9=4t−7=m9,9.
Thus, the neighborhoodM -polynomial of LSGt,s, for t≠1 and s=1, is(111)NMLSGt,s=∑i≤jmi,jxiyj,NMLSGt,s=t+8x4y4+2tx4y5+2tx5y8+2x8y8+4t−1x8y9+4t−7x9y9.
This is the required neighborhoodM -polynomial of LSGt,s, for t≠1 and s=1.Theorem 6.
LetLSGt,sbe the line graph of the subdivision graph of grapheneGt,s, witht=1ands>1; then, its neighborhoodM-polynomial is given by(112)NMLSGt,s=10x4y4+4x4y5+2s−2x5y5+4s−1x5y8+2s−1x8y8+4s−1x8y9+s−1x9y9.Proof.
The line graph of the subdivision graph of graphene has23st+2s+2t−1 vertices and 9st+4s+4t−5 edges. The edge set of LSGt,s, for t=1 and s>1, can be partitioned as follows:(113)E4,4=uv∈ELSGt,s:δu=4,δv=4=10=m4,4,E4,5=uv∈ELSGt,s:δu=4,δv=5=4=m4,5,E5,5=uv∈ELSGt,s:δu=5,δv=5=2s−2=m5,5,E5,8=uv∈ELSGt,s:δu=5,δv=8=4s−1=m5,8,E8,8=uv∈ELSGt,s:δu=8,δv=8=2s−1=m8,8,E8,9=uv∈ELSGt,s:δu=8,δv=9=4s−1=m8,9,E9,9=uv∈ELSGt,s:δu=9,δv=9=s−1=m9,9.
Thus, the neighborhoodM -polynomial of LSGt,s, for t=1 and s>1, is(114)NMLSGt,s=∑i≤jmi,jxiyj,NMLSGt,s=10x4y4+4x4y5+2s−2x5y5+4s−1x5y82s−1x8y8+4s−1x8y9+s−1x9y9.
This is the required neighborhoodM -polynomial of LSGt,s, for t=1 and s>1.
Now, we calculate neighborhood degree-based graphical indices of the line graph of the subdivision graph of graphene by using its neighborhoodM -polynomial.Corollary 4.
LetLSGt,sbe the line graph of the subdivision graph of graphene network, witht≠1ands>1; then, its neighborhood degree-based graphical indices are given by(1)
M1′=162st+28s+30t−126(2)
M2∗=729st−22s+19t−579(3)
FN∗=1458st−4s+62t−1198(4)
M2nm=1/819st−8s−5t+1+1/202s+t−21/18+s+t−2+2/25s−2+1/16t+6+1/10t+2+1/32s(5)
NRα=16αt+6+20α2t+2+25α2s−2+40α22s+t−2+64α2s+72α4s+t−2+81α9st−8s−5t+1(6)
ND3=13122st−2140s−866t−9926(7)
ND5=29st−8s−5t+1+89/202s+t−2+145/18s+t−2+4s−2+2t+6+41/10t+2+4s(8)
NH=1/99st−8s−5t+1+4/132s+t−2+8/17s+t−2+2/5s−2+1/4t+6+4/9t+2+1/4s(9)
NI=9/29st−8s−5t+1+80/132s+t−2+288/17s+t−2+5s−2+2t+6+40/9t+2+8s(10)
S=129.74633799st−8s−5t+1+96.168294522s+t−2+55296/125s+t−2+15625/256s−2+512/27t+6+46.64723032t+2+65536/343s.(11)
HM1G5LSGt,s=2916st−48s+100t−2356(12)
HM2G5LSGt,s=59049st−15910s−7813t−40675(13)
AG5LSGt,s=9st−8s−5t+1+1310/202s+t−2+172/6s+t−2+2s−2+t+6+95/10t+2+2s.(14)
GA5LSGt,s=9st−8s−5t+1+810/132s+t−2+482/17s+t−2+2s−2+t+6+85/9t+2+2sProof.
NM -Polynomial of LSGt,s, for t≠1 and s>1, is given by(115)NMLSGt,s=t+6x4y4+2t+2x4y5+2s−2x5y5+22s+t−2x5y8+2sx8y8+4s+t−2x8y9+9st−8s−5t+1x9y9.
Let(116)NMLSGt,s=fx,y,and then, we have(117)fx,y=NMLSGt,s=t+6x4y4+2t+2x4y5+2s−2x5y5+22s+t−2x5y8+2sx8y8+4s+t−2x8y9+9st−8s−5t+1x9y9.(1)
M1′ is defined as(118)M1′=Dx+DyNMG|x=y=1.Now, by using equation (117), we have(119)M1′=Dx+Dyfx,y|x=y=1,M1′=Dx+Dyt+6x4y4+2t+2x4y5+2s−2x5y5+22s+t−2x5y8+2sx8y8+4s+t−2x8y9+9st−8s−5t+1x9y9|x=y=1,M1′=4t+6x4y4+4t+6x4y4+8t+2x4y5+10t+2x4y5+10s−2x5y5+10s−2x5y5+102s+t−2x5y8+162s+t−2x5y8+16sx8y8+16sx8y8+32s+t−2x8y9+36s+t−2x8y9+99st−8s−5t+1x9y9+99st−8s−5t+1x9y9|x=y=1,M1′=162st+28s+30t−126.(2)
M2∗ is defined as(120)M2∗=DxDyNMG|x=y=1.Now, by using equation (117), we have(121)M2∗=DxDyfx,y|x=y=1,M2∗=DxDyt+6x4y4+2t+2x4y5+2s−2x5y5+22s+t−2x5y8+2sx8y8+4s+t−2x8y9+9st−8s−5t+1x9y9|x=y=1,M2∗=16t+6x4y4+40t+2x4y5+50s−2x5y5+802s+t−2x5y8+128sx8y8+288s+t−2x8y9+819st−8s−5t+1x9y9|x=y=1,M2∗=729st−22s+19t−579.(3)
FN∗ is defined as(122)FN∗=Dx2+Dy2NMG|x=y=1.Now, by using equation (117), we have(123)FN∗=Dx2+Dy2fx,y|x=y=1,FN∗=Dx2+Dy2t+6x4y4+2t+2x4y5+2s−2x5y5+22s+t−2x5y8+2sx8y8+4s+t−2x8y9+9st−8s−5t+1x9y9|x=y=1,FN∗=16t+6x4y4+16t+6x4y4+32t+2x4y5+50t+2x4y5+50s−2x5y5+50s−2x5y5+502s+t−2x5y8+1282s+t−2x5y8+128sx8y8+128sx8y8+256s+t−2x8y9+324s+t−2x8y9+819st−8s−5t+1x9y9+819st−8s−5t+1x9y9|x=y=1,FN∗=1458st−4s+62t−1198.(4)
M2nm is defined as(124)M2nm=SxSyNMG|x=y=1.Now, by using equation (117), we have(125)M2nm=SxSyfx,y|x=y=1,M2nm=SxSyt+6x4y4+2t+2x4y5+2s−2x5y5+22s+t−2x5y8+2sx8y8+4s+t−2x8y9+9st−8s−5t+1x9y9|x=y=1,M2nm=116t+6x4y4+220t+2x4y5+225s−2x5y5+2402s+t−2x5y8+264sx8y8+472s+t−2x8y9+1819st−8s−5t+1x9y9|x=y=1,M2nm=1819st−8s−5t+1+1202s+t−2+118s+t−2+225s−2+116t+6+110t+2+132s.(5)
NRα is defined as(126)NRα=DxαDyαNMG|x=y=1.Now, by using equation (117), we have(127)NRα=DxαDyαfx,y|x=y=1,NRα=DxαDyαt+6x4y4+2t+2x4y5+2s−2x5y5+22s+t−2x5y8+2sx8y8+4s+t−2x8y9+9st−8s−5t+1x9y9|x=y=1,NRα=16αt+6x4y4+20α2t+2x4y5+25α2s−2x5y5+40α22s+t−2x5y8+64α2sx8y8+72α4s+t−2x8y9+81α9st−8s−5t+1x9y9|x=y=1,NRα=16αt+6+20α2t+2+25α2s−2+40α22s+t−2+64α2s+72α4s+t−2+81α9st−8s−5t+1.(6)
ND3 is defined as(128)ND3=DxDyDx+DyNMG|x=y=1.Now, by using equation (117), we have(129)ND3=DxDyDx+Dyfx,y|x=y=1,ND3=DxDyDx+Dyt+6x4y4+2t+2x4y5+2s−2x5y5+22s+t−2x5y8+2sx8y8+4s+t−2x8y9+9st−8s−5t+1x9y9|x=y=1,ND3=DxDy4t+6x4y4+4t+6x4y4+8t+2x4y5+10t+2x4y5+10s−2x5y5+10s−2x5y5+102s+t−2x5y8+162s+t−2x5y8+16sx8y8+16sx8y8+32s+t−2x8y9+36s+t−2x8y9+99st−8s−5t+1x9y9+99st−8s−5t+1x9y9|x=y=1,ND3=128t+6x4y4+360t+2x4y5+500s−2x5y5+10402s+t−2x5y8+2048sx8y8+4896s+t−2x8y9+14589st−8s−5t+1x9y9|x=y=1,ND3=13122st−2140s−866t−9926.(7)
ND5 is defined as(130)ND5=DxSy+SxDyNMG|x=y=1.Now, by using equation (117), we have(131)ND5=DxSy+SxDyfx,y|x=y=1,ND5=DxSy+SxDyt+6x4y4+2t+2x4y5+2s−2x5y5+22s+t−2x5y8+2sx8y8+4s+t−2x8y9+9st−8s−5t+1x9y9|x=y=1,ND5=t+6x4y4+t+6x4y4+85t+2x4y5+104t+2x4y5+2s−2x5y5+2s−2x5y5+1082s+t−2x5y8+1652s+t−2x5y8+2sx8y8+2sx8y8+329s+t−2x8y9+368s+t−2x8y9+9st−8s−5t+1x9y9+9st−8s−5t+1x9y9|x=y=1,ND5=29st−8s−5t+1+89202s+t−2+14518s+t−2+4s−2+2t+6+4110t+2+4s.(8)
NH is defined as(132)NH=2SxJNMG|x=y=1.Now, by using equation (117), we have(133)NH=2SxJfx,y|x=y=1,NH=2SxJt+6x4y4+2t+2x4y5+2s−2x5y5+22s+t−2x5y8+2sx8y8+4s+t−2x8y9+9st−8s−5t+1x9y9|x=y=1,NH=28t+6x8+49t+2x9+410s−2x10+4132s+t−2x13+416sx16+817s+t−2x17+2189st−8s−5t+1x18|x=1,NH=199st−8s−5t+1+4132s+t−2+817s+t−2+25s−2+14t+6+49t+2+14s.(9)
NI is defined as(134)NI=SxJDxDyNMG|x=y=1.Now, by using equation (117), we have(135)NI=SxJDxDyfx,y|x=y=1,NI=SxJDxDyt+6x4y4+2t+2x4y5+2s−2x5y5+22s+t−2x5y8+2sx8y8+4s+t−2x8y9+9st−8s−5t+1x9y9|x=y=1,NI=SxJ16t+6x4y4+40t+2x4y5+50s−2x5y5+802s+t−2x5y8+128sx8y8+288s+t−2x8y9+819st−8s−5t+1x9y9|x=y=1,NI=168t+6x8+409t+2x9+5010s−2x10+80132s+t−2x13+12816sx16+28817s+t−2x17+81189st−8s−5t+1x18|x=1,NI=929st−8s−5t+1+80132s+t−2+28817s+t−2+5s−2+2t+6+409t+2+8s.(10)
S is defined as(136)S=Sx3Q−2JDx3Dy3NMG|x=y=1.Now, by using equation (117), we have(137)S=Sx3Q−2JDx3Dy3fx,y|x=y=1,S=Sx3Q−2JDx3Dy3t+6x4y4+2t+2x4y5+2s−2x5y5+22s+t−2x5y8+2sx8y8+4s+t−2x8y9+9st−8s−5t+1x9y9|x=y=1,S=4096216t+6x6+16000343t+2x7+31250512s−2x8+12800013312s+t−2x11+5242882744sx14+14929923375s+t−2x15+53144140969st−8s−5t+1x16|x=1,S=129.74633799st−8s−5t+1+96.168294522s+t−2+55296125s+t−2+15625256s−2+51227t+6+46.64723032t+2+65536343s.(11)
Fifth hyperM1 Zagreb index is defined as(138)HM1G5G=Dx2+Dy2+2DxDyNMG|x=y=1.Now, by using equation (117), we have(139)HM1G5LSGt,s=Dx2+Dy2+2DxDyfx,y|x=y=1,HM1G5LSGt,s=Dx2+Dy2+2DxDyt+6x4y4+2t+2x4y5+2s−2x5y5+22s+t−2x5y8+2sx8y8+4s+t−2x8y9+9st−8s−5t+1x9y9|x=y=1,HM1G5LSGt,s=16t+6x4y4+16t+6x4y4+32t+2x4y5+50t+2x4y5+50s−2x5y5+50s−2x5y5+502s+t−2x5y8+1282s+t−2x5y8+128sx8y8+128sx8y8+256s+t−2x8y9+324s+t−2x8y9+819st−8s−5t+1x9y9+819st−8s−5t+1x9y9+216t+6x4y4+40t+2x4y5+50s−2x5y5+802s+t−2x5y8+128sx8y8+288s+t−2x8y9+819st−8s−5t+1x9y9|x=y=1,HM1G5LSGt,s=2916st−48s+100t−2356.(12)
Fifth hyperM2 Zagreb index is defined as(140)HM2G5G=DxDyDxDyNMG|x=y=1.Now, by using equation (117), we have(141)HM2G5LSGt,s=DxDyDxDyfx,y|x=y=1,HM2G5LSGt,s=DxDyDxDyt+6x4y4+2t+2x4y5+2s−2x5y5+22s+t−2x5y8+2sx8y8+4s+t−2x8y9+9st−8s−5t+1x9y9|x=y=1,HM2G5LSGt,s=DxDy16t+6x4y4+40t+2x4y5+50s−2x5y5+802s+t−2x5y8+128sx8y8+288s+t−2x8y9+819st−8s−5t+1x9y9|x=y=1,HM2G5LSGt,s=256t+6x4y4+800t+2x4y5+1250s−2x5y5+32002s+t−2x5y8+8192sx8y8+20736s+t−2x8y9+65619st−8s−5t+1x9y9|x=y=1,HM2G5LSGt,s=59049st−15910s−7813t−40675.(13)
Fifth arithmetic-geometric index is defined as(142)AG5G=12Sx1/2Sy1/2Dx+DyNMG|x=y=1.Now, by using equation (117), we have(143)AG5LSGt,s=12Sx1/2S\y1/2Dx+Dyfx,y|x=y=1,AG5LSGt,s=12Sx1/2Sy1/2Dx+Dyt+6x4y4+2t+2x4y5+2s−2x5y5+22s+t−2x5y8+2sx8y8+4s+t−2x8y9+9st−8s−5t+1x9y9|x=y=1,AG5LSGt,s=12Sx1/2Sy1/24t+6x4y4+4t+6x4y4+8t+2x4y5+10t+2x4y5+10s−2x5y5+10s−2x5y5+102s+t−2x5y8+162s+t−2x5y8+16sx8y8+16sx8y8+32s+t−2x8y9+36s+t−2x8y9+99st−8s−5t+1x9y9+99st−8s−5t+1x9y9|x=y=1,AG5LSGt,s=8216t+6x4y4+18220t+2x4y5+20225s−2x5y5+262402s+t−2x5y8+32264sx8y8+68272s+t−2x8y9+182819st−8s−5t+1x9y9|x=y=1,AG5LSGt,s=9st−8s−5t+1+1310202s+t−2+1726s+t−2+2s−2+t+6+9510t+2+2s.(14)
Fifth geometric-arithmetic index is defined as(144)GA5G=2SxJDx1/2Dx1/2NMG|x=1.Now, by using equation (117), we have(145)GA5LSGt,s=2SxJDx1/2Dy1/2fx,y|x=1,GA5LSGt,s=2SxJDx1/2Dy1/2t+6x4y4+2t+2x4y5+2s−2x5y5+22s+t−2x5y8+2sx8y8+4s+t−2x8y9+9st−8s−5t+1x9y9|x=1,GA5LSGt,s=2168t+6x8+4209t+2x9+42510s−2x10+440132s+t−2x13+46416sx16+87217s+t−2x17+281189st−8s−5t+1x18|x=1,GA5LSGt,s=9st−8s−5t+1+810132s+t−248217s+t−2+2s−2+t+6859t+2+2s.Corollary 5.
LetLSGt,sbe the line graph of the subdivision graph of graphene network, witht≠1ands=1; then, its neighborhood degree-based graphical indices are given by(1)
M1′=192t−98(2)
M2∗=748t−599(3)
FN∗=1520t−1202(4)
M2nm=1/814t−7+1/16t+8+1/18t−1+3/20t+1/32(5)
NRα=16αt+8+20α2t+40α2t+64α2+72α4t−1+81α4t−7(6)
ND3=12256t−12030(7)
ND5=24t−7+2t+8+145/18t−1+171/20t+4(8)
NH=1/94t−7+1/4t+8+8/17t−1+88/117t+1/4(9)
NI=9/24t−7+2t+8+288/17t−1+1240/117t+8(10)
S=129.74633794t−7+512/27t+8+442.368t−1+142.8155248t+191.0670554(11)
HM1G5LSGt,s=3016t−2400(12)
HM2G5LSGt,s=51236t−56423(13)
AG5LSGt,s=4t−7+t+8+172/6t−1+4.067941659t+2(14)
GA5LSGt,s=4t−7+t+8+482/17t−1+3.93363300t+2Proof.
NM -Polynomial of LSGt,s, for t≠1 and s=1, is given by(146)NMLSGt,s=t+8x4y4+2tx4y5+2tx5y8+2x8y8+4t−1x8y9+4t−7x9y9.
Let(147)NMLSGt,s=fx,y,and then, we have(148)fx,y=NMLSGt,s=t+8x4y4+2tx4y5+2tx5y8+2x8y8+4t−1x8y9+4t−7x9y9.(1)
M1′ is defined as(149)M1′=Dx+DyNMG|x=y=1.Now, by using equation (148), we have(150)M1′=Dx+Dyfx,y|x=y=1,M1′=Dx+Dyt+8x4y4+2tx4y5+2tx5y8+2x8y8+4t−1x8y9+4t−7x9y9|x=y=1,M1′=4t+8x4y4+4t+8x4y4+8tx4y5+10tx4y5+10tx5y8+16tx5y8+16x8y8+16x8y8+32t−1x8y9+36t−1x8y9+94t−7x9y9+94t−7x9y9|x=y=1,M1′=192t−98.(2)
M2∗ is defined as(151)M2∗=DxDyNMG|x=y=1.Now, by using equation (148), we have(152)M2∗=DxDyfx,y|x=y=1,M2∗=DxDyt+8x4y4+2tx4y5+2tx5y8+2x8y8+4t−1x8y9+4t−7x9y9|x=y=1,M2∗=16t+8x4y4+40tx4y5+80tx5y8+128x8y8+288t−1x8y9+814t−7x9y9|x=y=1,M2∗=748t−599.(3)
FN∗ is defined as(153)FN∗=Dx2+Dy2NMG|x=y=1.Now, by using equation (148), we have(154)FN∗=Dx2+Dy2fx,y|x=y=1,FN∗=Dx2+Dy2t+8x4y4+2tx4y5+2tx5y8+2x8y8+4t−1x8y9+4t−7x9y9|x=y=1,FN∗=16t+8x4y4+16t+8x4y4+32tx4y5+50tx4y5+50tx5y8+128tx5y8+128x8y8+128x8y8+256t−1x8y9+324t−1x8y9+814t−7x9y9+814t−7x9y9|x=y=1,FN∗=1520t−1202.(4)
M2nm is defined as(155)M2nm=SxSyNMG|x=y=1.Now, by using equation (148), we have(156)M2nm=SxSyfx,y|x=y=1,M2nm=SxSyt+8x4y4+2tx4y5+2tx5y8+2x8y8+4t−1x8y9+4t−7x9y9|x=y=1,M2nm=116t+8x4y4+220tx4y5+240tx5y8+264x8y8+472t−1x8y9+1814t−7x9y9|x=y=1,M2nm=1814t−7+116t+8+118t−1+320t+132.(5)
NRα is defined as(157)NRα=DxαDyαNMG|x=y=1.Now, by using equation (148), we have(158)NRα=DxαDyαfx,y|x=y=1,NRα=DxαDyαt+8x4y4+2tx4y5+2tx5y8+2x8y8+4t−1x8y9+4t−7x9y9|x=y=1,NRα=16αt+8x4y4+20α2tx4y5+40α2tx5y8+64α2x8y8+72α4t−1x8y9+81α4t−7x9y9|x=y=1,NRα=16αt+8+20α2t+40α2t+64α2+72α4t−1+81α4t−7.(6)
ND3 is defined as(159)ND3=DxDyDx+DyNMG|x=y=1.Now, by using equation (148), we have(160)ND3=DxDyDx+Dyfx,y|x=y=1,ND3=DxDyDx+Dyt+8x4y4+2tx4y5+2tx5y8+2x8y8+4t−1x8y9+4t−7x9y9|x=y=1,ND3=DxDy4t+8x4y4+4t+8x4y4+8tx4y5+10tx4y5+10tx5y8+16tx5y8+16x8y8+16x8y8+32t−1x8y9+36t−1x8y9+94t−7x9y9+94t−7x9y9|x=y=1,ND3=128t+8x4y4+360tx4y5+1040tx5y8+2048x8y8+4896t−1x8y9+14584t−7x9y9|x=y=1,ND3=12256t−12030.(7)
ND5 is defined as(161)ND5=DxSy+SxDyNMG|x=y=1.Now, by using equation (148), we have(162)ND5=DxSy+SxDyfx,y|x=y=1,ND5=DxSy+SxDyt+8x4y4+2tx4y5+2tx5y8+2x8y8+4t−1x8y9+4t−7x9y9|x=y=1,ND5=t+8x4y4+t+8x4y4+85tx4y5+104tx4y5+108tx5y8+165tx5y8+2x8y8+2x8y8+329t−1x8y9+368t−1x8y9+4t−7x9y9+4t−7x9y9|x=y=1,ND5=24t−7+2t+8+14518t−1+17120t+4.(8)
NH is defined as(163)NH=2SxJNMG|x=y=1.Now, by using equation (148), we have(164)NH=2SxJfx,y|x=y=1,NH=2SxJt+8x4y4+2tx4y5+2tx5y8+2x8y8+4t−1x8y9+4t−7x9y9|x=y=1,NH=28t+8x8+49tx9+413tx13+416x16+817t−1x17+2184t−7x18|x=1,NH=194t−7+14t+8+817t−1+88117t+14.(9)
NI is defined as(165)NI=SxJDxDyNMG|x=y=1.Now, by using equation (148), we have(166)NI=SxJDxDyfx,y|x=y=1,NI=SxJDxDyt+8x4y4+2tx4y5+2tx5y8+2x8y8+4t−1x8y9+4t−7x9y9|x=y=1,NI=SxJ16t+8x4y4+40tx4y5+80tx5y8+128x8y8+288t−1x8y9+814t−7x9y9|x=y=1,NI=168t+8x8+409tx9+8013tx13+12816x16+28817t−1x17+81184t−7x18|x=1,NI=924t−7+2t+8+28817t−1+1240117t+8.(10)
S is defined as(167)S=Sx3Q−2JDx3Dy3NMG|x=y=1.Now, by using equation (148), we have(168)S=Sx3Q−2JDx3Dy3fx,y|x=y=1,S=Sx3Q−2JDx3Dy3t+8x4y4+2tx4y5+2tx5y8+2x8y8+4t−1x8y9+4t−7x9y9|x=y=1,S=4096216t+8x6+16000343tx7+1280001331tx11+5242882744x14+14929923375t−1x15+53144140964t−7x16|x=1,S=129.74633794t−7+51227t+8+442.368t−1+142.8155248t+191.0670554.(11)
Fifth hyperM1 Zagreb index is defined as(169)HM1G5G=Dx2+Dy2+2DxDyNMG|x=y=1.Now, by using equation (148), we have(170)HM1G5LSGt,s=Dx2+Dy2+2DxDyfx,y|x=y=1,HM1G5LSGt,s=Dx2+Dy2+2DxDyt+8x4y4+2tx4y5+2tx5y8+2x8y8+4t−1x8y9+4t−7x9y9|x=y=1,HM1G5LSGt,s=16t+8x4y4+16t+8x4y4+32tx4y5+50tx4y5+50tx5y8+128tx5y8+128x8y8+128x8y8+256t−1x8y9+324t−1x8y9+814t−7x9y9+814t−7x9y9+216t+8x4y4+40tx4y5+80tx5y8+128x8y8+288t−1x8y9+814t−7x9y9|x=y=1,HM1G5LSGt,s=3016t−2400.(12)
Fifth hyperM2 Zagreb index is defined as(171)HM2G5G=DxDyDxDyNMG|x=y=1.Now, by using equation (148), we have(172)HM2G5LSGt,s=DxDyDxDyfx,y|x=y=1,HM2G5LSGt,s=DxDyDxDyt+8x4y4+2tx4y5+2tx5y8+2x8y8+4t−1x8y9+4t−7x9y9|x=y=1,HM2G5LSGt,s=DxDy16t+8x4y4+40tx4y5+80tx5y8+128x8y8+288t−1x8y9+814t−7x9y9|x=y=1,HM2G5LSGt,s=256t+8x4y4+800tx4y5+3200tx5y8+8192x8y8+20736t−1x8y9+65614t−7x9y9|x=y=1,HM2G5LSGt,s=51236t−56423.(13)
Fifth arithmetic-geometric index is defined as(173)AG5G=12Sx1/2Sy1/2Dx+DyNMG|x=y=1.Now, by using equation (148), we have(174)AG5LSGt,s=12Sx1/2Sy1/2Dx+Dyfx,y|x=y=1,AG5LSGt,s=12Sx1/2Sy1/2Dx+Dyt+8x4y4+2tx4y5+2tx5y8+2x8y8+4t−1x8y9+4t−7x9y9|x=y=1,AG5LSGt,s=12Sx1/2Sy1/24t+8x4y4+4t+8x4y4+8tx4y5+10tx4y5+10tx5y8+16tx5y8+16x8y8+16x8y8+32t−1x8y9+36t−1x8y9+94t−7x9y9+94t−7x9y9|x=y=1,AG5LSGt,s=8216t+8x4y4+18220tx4y5+26240tx5y8+32264x8y8+68272t−1x8y9+182814t−7x9y9|x=y=1,AG5LSGt,s=4t−7+t+8+1726t−1+4.067941659t+2.(14)
Fifth geometric-arithmetic index is defined as(175)GA5G=2SxJDx1/2Dy1/2NMG|x=1.Now, by using Equation (16), we have(176)GA5LSGt,s=2SxJDx1/2Dy1/2fx,y|x=1,GA5LSGt,s=2SxJDx1/2Dy1/2t+8x4y4+2tx4y5+2tx5y8+2x8y8+4t−1x8y9+4t−7x9y9|x=1,GA5LSGt,s=2168t+8x8+4209tx9+44013tx13+46416x16+87217t−1x17+281184t−7x18|x=1,GA5LSGt,s=4t−7+t+8+48217t−1+3.93363300t+2.Corollary 6.
LetLSGt,sbe the line graph of the subdivision graph of graphene network, witht=1ands>1; then, its neighborhood degree-based graphical indices are given by(1)
M1′=190s−94(2)
M2∗=704s−517(3)
FN∗=1454s−1070(4)
M2nm=2/25s−2+2581/12960s−1+33/40(5)
NRα=16α10+20α4+25α2s−2+40α4s−1+64α2s−1+72α4s−1+81αs−1.(6)
ND3=10982s−9482(7)
ND5=4s−2+1033/45s−1+141/5(8)
NH=2/5s−2+1.447083962s−1+61/18(9)
NI=5s−2+18453/442s−1+260/9(10)
S=15625/256s−2+955.5179823s−1+282.9240903(11)
HM1G5LSGt,s=2862s−2104(12)
HM2G5LSGt,s=43139s−40229(13)
AG5LSGt,s=2s−2+11.11789938s−1+14.02492236(14)
GA5LSGt,s=2s−2+10.88510763s−1+13.97523196Proof.
NM-Polynomial of LSGt,s, for t=1 and s>1, is given by(177)NMLSGt,s=10x4y4+4x4y5+2s−2x5y5+4s−1x5y8+2s−1x8y8+4s−1x8y9+s−1x9y9.
Let(178)NMLSGt,s=fx,y,and then, we have(179)fx,y=NMLSGt,s=10x4y4+4x4y5+2s−2x5y5+4s−1x5y8+2s−1x8y8+4s−1x8y9+s−1x9y9.(1)
M1′ is defined as(180)M1′=Dx+DyNMG|x=y=1.Now, by using equation (179), we have(181)M1′=Dx+Dyfx,y|x=y=1,M1′=Dx+Dy10x4y4+4x4y5+2s−2x5y5+4s−1x5y8+2s−1x8y8+4s−1x8y9+s−1x9y9|x=y=1,M1′=40x4y4+40x4y4+16x4y5+20x4y5+10s−2x5y5+10s−2x5y5+20s−1x5y8+32s−1x5y8+16s−1x8y8+16s−1x8y8+32s−1x8y9+36s−1x8y9+9s−1x9y9+9s−1x9y9|x=y=1,M1′=190s−94.(2)
M2∗ is defined as(182)M2∗=DxDyNMG|x=y=1.Now, by using equation (179), we have(183)M2∗=DxDyfx,y|x=y=1,M2∗=DxDy10x4y4+4x4y5+2s−2x5y5+4s−1x5y8+2s−1x8y8+4s−1x8y9+s−1x9y9|x=y=1,M2∗=160x4y4+80x4y5+50s−2x5y5+160s−1x5y8+128s−1x8y8+288s−1x8y9+81s−1x9y9|x=y=1,M2∗=704s−517.(3)
FN∗ is defined as(184)FN∗=Dx2+Dy2NMG|x=y=1.Now, by using equation (179), we have(185)FN∗=Dx2+Dy2fx,y|x=y=1,FN∗=Dx2+Dy210x4y4+4x4y5+2s−2x5y5+4s−1x5y8+2s−1x8y8+4s−1x8y9+s−1x9y9|x=y=1,FN∗=160x4y4+160x4y4+64x4y5+100x4y5+50s−2x5y5+50s−2x5y5+100s−1x5y8+256s−1x5y8+128s−1x8y8+128s−1x8y8+256s−1x8y9+324s−1x8y9+81s−1x9y9+81s−1x9y9|x=y=1,FN∗=1454s−1070.(4)
M2nm is defined as(186)M2nm=SxSyNMG|x=y=1.Now, by using equation (179), we have(187)M2nm=SxSyfx,y|x=y=1,M2nm=SxSy10x4y4+4x4y5+2s−2x5y5+4s−1x5y8+2s−1x8y8+4s−1x8y9+s−1x9y9|x=y=1,M2nm=1016x4y4+420x4y5+225s−2x5y5+440s−1x5y8+264s−1x8y8+472s−1x8y9+181s−1x9y9|x=y=1,M2nm=225s−2+258112960s−1+3340.(5)
NRα is defined as(188)NRα=DxαDyαNMG|x=y=1.Now, by using equation (179), we have(189)NRα=DxαDyαfx,y|x=y=1,NRα=DxαDyα10x4y4+4x4y5+2s−2x5y5+4s−1x5y8+2s−1x8y8+4s−1x8y9+s−1x9y9|x=y=1,NRα=16α10x4y4+20α4x4y5+25α2s−2x5y5+40α4s−1x5y8+64α2s−1x8y8+72α4s−1x8y9+81αs−1x9y9|x=y=1,NRα=16α10+20α4+25α2s−2+40α4s−1+64α2s−1+72α4s−1+81αs−1.(6)
ND3 is defined as(190)ND3=DxDyDx+DyNMG|x=y=1.Now, by using equation (179), we have(191)ND3=DxDyDx+Dyfx,y|x=y=1,ND3=DxDyDx+Dy10x4y4+4x4y5+2s−2x5y5+4s−1x5y8+2s−1x8y8+4s−1x8y9+s−1x9y9|x=y=1,ND3=DxDy40x4y4+40x4y4+16x4y5+20x4y5+10s−2x5y5+10s−2x5y5+20s−1x5y8+32s−1x5y8+16s−1x8y8+16s−1x8y8+32s−1x8y9+36s−1x8y9+9s−1x9y9+9s−1x9y9|x=y=1,ND3=1280x4y4+720x4y5+500s−2x5y5+2080s−1x5y8+2048s−1x8y8+4896s−1x8y9+1458s−1x9y9|x=y=1,ND3=10982s−9482.(7)
ND5 is defined as(192)ND5=DxSy+SxDyNMG|x=y=1.Now, by using equation (179), we have(193)ND5=DxSy+SxDyfx,y|x=y=1,ND5=DxSy+SxDy10x4y4+4x4y5+2s−2x5y5+4s−1x5y8+2s−1x8y8+4s−1x8y9+s−1x9y9|x=y=1,ND5=10x4y4+10x4y4+165x4y5+204x4y5+2s−2x5y5+2s−2x5y5+208s−1x5y8+325s−1x5y8+2s−1x8y8+2s−1x8y8+329s−1x8y9+368s−1x8y9+s−1x9y9+s−1x9y9|x=y=1,ND5=4s−2+103345s−1+1415.(8)
NH is defined as(194)NH=2SxJNMG|x=y=1.Now, by using equation (179), we have(195)NH=2SxJfx,y|x=y=1,NH=2SxJ10x4y4+4x4y5+2s−2x5y5+4s−1x5y8+2s−1x8y8+4s−1x8y9+s−1x9y9|x=y=1,NH=208x8+89x9+410s−2x10+813s−1x13+416s−1x16+817s−1x17+218s−1x18|x=1,NH=25s−2+1.447083962s−1+6118.(9)
NI is defined as(196)NI=SxJDxDyNMG|x=y=1.Now, by using equation (179), we have(197)NI=SxJDxDyfx,y|x=y=1,NI=SxJDxDy10x4y4+4x4y5+2s−2x5y5+4s−1x5y8+2s−1x8y8+4s−1x8y9+s−1x9y9|x=y=1,NI=SxJ160x4y4+80x4y5+50s−2x5y5+160s−1x5y8+128s−1x8y8+288s−1x8y9+81s−1x9y9|x=y=1,NI=1608x8+809x9+5010s−2x10+16013s−1x13+12816s−1x16+28817s−1x17+8118s−1x18|x=1,NI=5s−2+18453442s−1+2609.(10)
S is defined as(198)S=Sx3Q−2JDx3Dy3NMG|x=y=1.Now, by using equation (179), we have(199)S=Sx3Q−2JDx3Dy3fx,y|x=y=1,S=Sx3Q−2JDx3Dy310x4y4+4x4y5+2s−2x5y5+4s−1x5y8+2s−1x8y8+4s−1x8y9+s−1x9y9|x=y=1,S=40960216x6+32000343x7+31250512s−2x8+2560001331s−1x11+5242882744s−1x14+14929923375s−1x15+5314414096s−1x16|x=1,S=15625256s−2+955.5179823s−1+282.9240903.(11)
Fifth hyperM1 Zagreb index is defined as(200)HM1G5G=Dx2+Dy2+2DxDyNMG|x=y=1.Now, by using equation (179), we have(201)HM1G5LSGt,s=Dx2+Dy2+2DxDyfx,y|x=y=1,HM1G5LSGt,s=Dx2+Dy2+2DxDy10x4y4+4x4y5+2s−2x5y5+4s−1x5y8+2s−1x8y8+4s−1x8y9+s−1x9y9|x=y=1,HM1G5LSGt,s=160x4y4+160x4y4+64x4y5+100x4y5+50s−2x5y5+50s−2x5y5+100s−1x5y8+256s−1x5y8+128s−1x8y8+128s−1x8y8+256s−1x8y9+324s−1x8y9+81s−1x9y9+81s−1x9y9+2160x4y4+80x4y5+50s−2x5y5+160s−1x5y8+128s−1x8y8+288s−1x8y9+81s−1x9y9|x=y=1,HM1G5LSGt,s=2862s−2104.(12)
Fifth hyperM2 Zagreb index is defined as(202)HM2G5G=DxDyDxDyNMG|x=y=1.Now, by using equation (179), we have(203)HM2G5LSGt,s=DxDyDxDyfx,y|x=y=1,HM2G5LSGt,s=DxDyDxDy10x4y4+4x4y5+2s−2x5y5+4s−1x5y8+2s−1x8y8+4s−1x8y9+s−1x9y9|x=y=1,HM2G5LSGt,s=DxDy160x4y4+80x4y5+50s−2x5y5+160s−1x5y8+128s−1x8y8+288s−1x8y9+81s−1x9y9|x=y=1,HM2G5LSGt,s=2560x4y4+1600x4y5+1250s−2x5y5+6400s−1x5y8+8192s−1x8y8+20736s−1x8y9+6561s−1x9y9|x=y=1,HM2G5LSGt,s=43139s−40229.(13)
Fifth arithmetic-geometric index is defined as(204)AG5G=12Sx1/2Sy1/2Dx+DyNMG|x=y=1.Now, by using equation (179), we have(205)AG5LSGt,s=12Sx1/2Sy1/2Dx+Dyfx,y|x=y=1,AG5LSGt,s=12Sx1/2Sy1/2Dx+Dy10x4y4+4x4y5+2s−2x5y5+4s−1x5y8+2s−1x8y8+4s−1x8y9+s−1x9y9|x=y=1,AG5LSGt,s=12Sx1/2Sy1/240x4y4+40x4y4+16x4y5+20x4y5+10s−2x5y5+10s−2x5y5+20s−1x5y8+32s−1x5y8+16s−1x8y8+16s−1x8y8+32s−1x8y9+36s−1x8y9+9s−1x9y9+9s−1x9y9|x=y=1,AG5LSGt,s=80216x4y4+36220x4y5+20225s−2x5y5+52240s−1x5y8+32264s−1x8y8+6872s−1x8y9+18281s−1x9y9|x=y=1,AG5LSGt,s=2s−2+11.11789938s−1+14.02492236.(14)
Fifth geometric-arithmetic index is defined as(206)GA5G=2SxJDx1/2Dy1/2NMG|x=1.Now, by using equation (179), we have(207)GA5LSGt,s=2SxJDx1/2Dy1/2fx,y|x=1,GA5LSGt,s=2SxJDx1/2Dy1/210x4y4+4x4y5+2s−2x5y5+4s−1x5y8+2s−1x8y8+4s−1x8y9+s−1x9y9|x=1,GA5LSGt,s=20168x8+8209x9+42510s−2x10+84013s−1x13+46416s−1x16+87217s−1x17+28118s−1x18|x=1,GA5LSGt,s=2s−2+10.88510763s−1+13.97523196.


## 4.1. Line Graph of the Subdivision Graph of Graphene Networks

The subdivision graph of grapheneGt,s is denoted by SGt,s, and the line graph of the subdivision graph of graphene is denoted by LSGt,s. The line graph of the subdivision graph of graphene has 23st+2s+2t−1 vertices and 9st+4s+4t−5 edges [34]. The subdivision graph of Gt,s and the line graph of the subdivision graph of Gt,s are shown in Figures 2 and 3 respectively.Figure 2
Subdivision graph ofGt,s.Figure 3
Line graph of the subdivision graph ofGt,s.Theorem 4.
LetLSGt,sbe the line graph of the subdivision graph of grapheneGt,s, witht≠1ands>1; then, its neighborhoodM-polynomial is given by(106)NMLSGt,s=t+6x4y4+2t+2x4y5+2s−2x5y5+22s+t−2x5y8+2sx8y8+4s+t−2x8y9+9st−8s−5t+1x9y9.Proof.
The line graph of the subdivision graph of graphene has23st+2s+2t−1 vertices and 9st+4s+4t−5 edges. The edge set of LSGt,s, for t≠1 and s>1, can be partitioned as follows:(107)E4,4=uv∈ELSGt,s:δu=4,δv=4=t+6=m4,4,E4,5=uv∈ELSGt,s:δu=4,δv=5=2t+2=m4,5,E5,5=uv∈ELSGt,s:δu=5,δv=5=2s−2=m5,5,E5,8=uv∈ELSGt,s:δu=5,δv=8=22s+t−2=m5,8,E8,8=uv∈ELSGt,s:δu=8,δv=8=2s=m8,8.E8,9=uv∈ELSGt,s:δu=8,δv=9=4s+t−2=m8,9,E9,9=uv∈ELSGt,s:δu=9,δv=9=9st−8s−5t+1=m9,9,
Thus, the neighborhoodM -polynomial of LSGt,s, for t≠1 and s>1, is(108)NMLSGt,s=∑i≤jmi,jxiyjm,NMLSGt,s=t+6x4y4+2t+2x4y5+2s−2x5y5+22s+t−2x5y8+2sx8y8+4s+t−2x8y9+9st−8s−5t+1x9y9.
This is the required neighborhoodM -polynomial of LSGt,s, for t≠1 and s>1.Theorem 5.
LetLSGt,sbe the line graph of the subdivision graph of grapheneGt,s, witht≠1ands=1; then, its neighborhoodM-polynomial is given by(109)NMLSGt,s=t+8x4y4+2tx4y5+2tx5y8+2x8y8+4t−1x8y9+4t−7x9y9.Proof.
The line graph of the subdivision graph of graphene has23st+2s+2t−1 vertices and 9st+4s+4t−5 edges. The edge set of LSGt,s, for t≠1 and s=1, can be partitioned as follows:'(110)E4,4=uv∈ELSGt,s:δu=4,δv=4=t+8=m4,4,E4,5=uv∈ELSGt,s:δu=4,δv=5=2t=m4,5,E5,8=uv∈ELSGt,s:δu=5,δv=8=2t=m5,8,E8,8=uv∈ELSGt,s:δu=8,δv=8=2=m8,8,E8,9=uv∈ELSGt,s:δu=8,δv=9=4t−1=m8,9,E9,9=uv∈ELSGt,s:δu=9,δv=9=4t−7=m9,9.
Thus, the neighborhoodM -polynomial of LSGt,s, for t≠1 and s=1, is(111)NMLSGt,s=∑i≤jmi,jxiyj,NMLSGt,s=t+8x4y4+2tx4y5+2tx5y8+2x8y8+4t−1x8y9+4t−7x9y9.
This is the required neighborhoodM -polynomial of LSGt,s, for t≠1 and s=1.Theorem 6.
LetLSGt,sbe the line graph of the subdivision graph of grapheneGt,s, witht=1ands>1; then, its neighborhoodM-polynomial is given by(112)NMLSGt,s=10x4y4+4x4y5+2s−2x5y5+4s−1x5y8+2s−1x8y8+4s−1x8y9+s−1x9y9.Proof.
The line graph of the subdivision graph of graphene has23st+2s+2t−1 vertices and 9st+4s+4t−5 edges. The edge set of LSGt,s, for t=1 and s>1, can be partitioned as follows:(113)E4,4=uv∈ELSGt,s:δu=4,δv=4=10=m4,4,E4,5=uv∈ELSGt,s:δu=4,δv=5=4=m4,5,E5,5=uv∈ELSGt,s:δu=5,δv=5=2s−2=m5,5,E5,8=uv∈ELSGt,s:δu=5,δv=8=4s−1=m5,8,E8,8=uv∈ELSGt,s:δu=8,δv=8=2s−1=m8,8,E8,9=uv∈ELSGt,s:δu=8,δv=9=4s−1=m8,9,E9,9=uv∈ELSGt,s:δu=9,δv=9=s−1=m9,9.
Thus, the neighborhoodM -polynomial of LSGt,s, for t=1 and s>1, is(114)NMLSGt,s=∑i≤jmi,jxiyj,NMLSGt,s=10x4y4+4x4y5+2s−2x5y5+4s−1x5y82s−1x8y8+4s−1x8y9+s−1x9y9.
This is the required neighborhoodM -polynomial of LSGt,s, for t=1 and s>1.
Now, we calculate neighborhood degree-based graphical indices of the line graph of the subdivision graph of graphene by using its neighborhoodM -polynomial.Corollary 4.
LetLSGt,sbe the line graph of the subdivision graph of graphene network, witht≠1ands>1; then, its neighborhood degree-based graphical indices are given by(1)
M1′=162st+28s+30t−126(2)
M2∗=729st−22s+19t−579(3)
FN∗=1458st−4s+62t−1198(4)
M2nm=1/819st−8s−5t+1+1/202s+t−21/18+s+t−2+2/25s−2+1/16t+6+1/10t+2+1/32s(5)
NRα=16αt+6+20α2t+2+25α2s−2+40α22s+t−2+64α2s+72α4s+t−2+81α9st−8s−5t+1(6)
ND3=13122st−2140s−866t−9926(7)
ND5=29st−8s−5t+1+89/202s+t−2+145/18s+t−2+4s−2+2t+6+41/10t+2+4s(8)
NH=1/99st−8s−5t+1+4/132s+t−2+8/17s+t−2+2/5s−2+1/4t+6+4/9t+2+1/4s(9)
NI=9/29st−8s−5t+1+80/132s+t−2+288/17s+t−2+5s−2+2t+6+40/9t+2+8s(10)
S=129.74633799st−8s−5t+1+96.168294522s+t−2+55296/125s+t−2+15625/256s−2+512/27t+6+46.64723032t+2+65536/343s.(11)
HM1G5LSGt,s=2916st−48s+100t−2356(12)
HM2G5LSGt,s=59049st−15910s−7813t−40675(13)
AG5LSGt,s=9st−8s−5t+1+1310/202s+t−2+172/6s+t−2+2s−2+t+6+95/10t+2+2s.(14)
GA5LSGt,s=9st−8s−5t+1+810/132s+t−2+482/17s+t−2+2s−2+t+6+85/9t+2+2sProof.
NM -Polynomial of LSGt,s, for t≠1 and s>1, is given by(115)NMLSGt,s=t+6x4y4+2t+2x4y5+2s−2x5y5+22s+t−2x5y8+2sx8y8+4s+t−2x8y9+9st−8s−5t+1x9y9.
Let(116)NMLSGt,s=fx,y,and then, we have(117)fx,y=NMLSGt,s=t+6x4y4+2t+2x4y5+2s−2x5y5+22s+t−2x5y8+2sx8y8+4s+t−2x8y9+9st−8s−5t+1x9y9.(1)
M1′ is defined as(118)M1′=Dx+DyNMG|x=y=1.Now, by using equation (117), we have(119)M1′=Dx+Dyfx,y|x=y=1,M1′=Dx+Dyt+6x4y4+2t+2x4y5+2s−2x5y5+22s+t−2x5y8+2sx8y8+4s+t−2x8y9+9st−8s−5t+1x9y9|x=y=1,M1′=4t+6x4y4+4t+6x4y4+8t+2x4y5+10t+2x4y5+10s−2x5y5+10s−2x5y5+102s+t−2x5y8+162s+t−2x5y8+16sx8y8+16sx8y8+32s+t−2x8y9+36s+t−2x8y9+99st−8s−5t+1x9y9+99st−8s−5t+1x9y9|x=y=1,M1′=162st+28s+30t−126.(2)
M2∗ is defined as(120)M2∗=DxDyNMG|x=y=1.Now, by using equation (117), we have(121)M2∗=DxDyfx,y|x=y=1,M2∗=DxDyt+6x4y4+2t+2x4y5+2s−2x5y5+22s+t−2x5y8+2sx8y8+4s+t−2x8y9+9st−8s−5t+1x9y9|x=y=1,M2∗=16t+6x4y4+40t+2x4y5+50s−2x5y5+802s+t−2x5y8+128sx8y8+288s+t−2x8y9+819st−8s−5t+1x9y9|x=y=1,M2∗=729st−22s+19t−579.(3)
FN∗ is defined as(122)FN∗=Dx2+Dy2NMG|x=y=1.Now, by using equation (117), we have(123)FN∗=Dx2+Dy2fx,y|x=y=1,FN∗=Dx2+Dy2t+6x4y4+2t+2x4y5+2s−2x5y5+22s+t−2x5y8+2sx8y8+4s+t−2x8y9+9st−8s−5t+1x9y9|x=y=1,FN∗=16t+6x4y4+16t+6x4y4+32t+2x4y5+50t+2x4y5+50s−2x5y5+50s−2x5y5+502s+t−2x5y8+1282s+t−2x5y8+128sx8y8+128sx8y8+256s+t−2x8y9+324s+t−2x8y9+819st−8s−5t+1x9y9+819st−8s−5t+1x9y9|x=y=1,FN∗=1458st−4s+62t−1198.(4)
M2nm is defined as(124)M2nm=SxSyNMG|x=y=1.Now, by using equation (117), we have(125)M2nm=SxSyfx,y|x=y=1,M2nm=SxSyt+6x4y4+2t+2x4y5+2s−2x5y5+22s+t−2x5y8+2sx8y8+4s+t−2x8y9+9st−8s−5t+1x9y9|x=y=1,M2nm=116t+6x4y4+220t+2x4y5+225s−2x5y5+2402s+t−2x5y8+264sx8y8+472s+t−2x8y9+1819st−8s−5t+1x9y9|x=y=1,M2nm=1819st−8s−5t+1+1202s+t−2+118s+t−2+225s−2+116t+6+110t+2+132s.(5)
NRα is defined as(126)NRα=DxαDyαNMG|x=y=1.Now, by using equation (117), we have(127)NRα=DxαDyαfx,y|x=y=1,NRα=DxαDyαt+6x4y4+2t+2x4y5+2s−2x5y5+22s+t−2x5y8+2sx8y8+4s+t−2x8y9+9st−8s−5t+1x9y9|x=y=1,NRα=16αt+6x4y4+20α2t+2x4y5+25α2s−2x5y5+40α22s+t−2x5y8+64α2sx8y8+72α4s+t−2x8y9+81α9st−8s−5t+1x9y9|x=y=1,NRα=16αt+6+20α2t+2+25α2s−2+40α22s+t−2+64α2s+72α4s+t−2+81α9st−8s−5t+1.(6)
ND3 is defined as(128)ND3=DxDyDx+DyNMG|x=y=1.Now, by using equation (117), we have(129)ND3=DxDyDx+Dyfx,y|x=y=1,ND3=DxDyDx+Dyt+6x4y4+2t+2x4y5+2s−2x5y5+22s+t−2x5y8+2sx8y8+4s+t−2x8y9+9st−8s−5t+1x9y9|x=y=1,ND3=DxDy4t+6x4y4+4t+6x4y4+8t+2x4y5+10t+2x4y5+10s−2x5y5+10s−2x5y5+102s+t−2x5y8+162s+t−2x5y8+16sx8y8+16sx8y8+32s+t−2x8y9+36s+t−2x8y9+99st−8s−5t+1x9y9+99st−8s−5t+1x9y9|x=y=1,ND3=128t+6x4y4+360t+2x4y5+500s−2x5y5+10402s+t−2x5y8+2048sx8y8+4896s+t−2x8y9+14589st−8s−5t+1x9y9|x=y=1,ND3=13122st−2140s−866t−9926.(7)
ND5 is defined as(130)ND5=DxSy+SxDyNMG|x=y=1.Now, by using equation (117), we have(131)ND5=DxSy+SxDyfx,y|x=y=1,ND5=DxSy+SxDyt+6x4y4+2t+2x4y5+2s−2x5y5+22s+t−2x5y8+2sx8y8+4s+t−2x8y9+9st−8s−5t+1x9y9|x=y=1,ND5=t+6x4y4+t+6x4y4+85t+2x4y5+104t+2x4y5+2s−2x5y5+2s−2x5y5+1082s+t−2x5y8+1652s+t−2x5y8+2sx8y8+2sx8y8+329s+t−2x8y9+368s+t−2x8y9+9st−8s−5t+1x9y9+9st−8s−5t+1x9y9|x=y=1,ND5=29st−8s−5t+1+89202s+t−2+14518s+t−2+4s−2+2t+6+4110t+2+4s.(8)
NH is defined as(132)NH=2SxJNMG|x=y=1.Now, by using equation (117), we have(133)NH=2SxJfx,y|x=y=1,NH=2SxJt+6x4y4+2t+2x4y5+2s−2x5y5+22s+t−2x5y8+2sx8y8+4s+t−2x8y9+9st−8s−5t+1x9y9|x=y=1,NH=28t+6x8+49t+2x9+410s−2x10+4132s+t−2x13+416sx16+817s+t−2x17+2189st−8s−5t+1x18|x=1,NH=199st−8s−5t+1+4132s+t−2+817s+t−2+25s−2+14t+6+49t+2+14s.(9)
NI is defined as(134)NI=SxJDxDyNMG|x=y=1.Now, by using equation (117), we have(135)NI=SxJDxDyfx,y|x=y=1,NI=SxJDxDyt+6x4y4+2t+2x4y5+2s−2x5y5+22s+t−2x5y8+2sx8y8+4s+t−2x8y9+9st−8s−5t+1x9y9|x=y=1,NI=SxJ16t+6x4y4+40t+2x4y5+50s−2x5y5+802s+t−2x5y8+128sx8y8+288s+t−2x8y9+819st−8s−5t+1x9y9|x=y=1,NI=168t+6x8+409t+2x9+5010s−2x10+80132s+t−2x13+12816sx16+28817s+t−2x17+81189st−8s−5t+1x18|x=1,NI=929st−8s−5t+1+80132s+t−2+28817s+t−2+5s−2+2t+6+409t+2+8s.(10)
S is defined as(136)S=Sx3Q−2JDx3Dy3NMG|x=y=1.Now, by using equation (117), we have(137)S=Sx3Q−2JDx3Dy3fx,y|x=y=1,S=Sx3Q−2JDx3Dy3t+6x4y4+2t+2x4y5+2s−2x5y5+22s+t−2x5y8+2sx8y8+4s+t−2x8y9+9st−8s−5t+1x9y9|x=y=1,S=4096216t+6x6+16000343t+2x7+31250512s−2x8+12800013312s+t−2x11+5242882744sx14+14929923375s+t−2x15+53144140969st−8s−5t+1x16|x=1,S=129.74633799st−8s−5t+1+96.168294522s+t−2+55296125s+t−2+15625256s−2+51227t+6+46.64723032t+2+65536343s.(11)
Fifth hyperM1 Zagreb index is defined as(138)HM1G5G=Dx2+Dy2+2DxDyNMG|x=y=1.Now, by using equation (117), we have(139)HM1G5LSGt,s=Dx2+Dy2+2DxDyfx,y|x=y=1,HM1G5LSGt,s=Dx2+Dy2+2DxDyt+6x4y4+2t+2x4y5+2s−2x5y5+22s+t−2x5y8+2sx8y8+4s+t−2x8y9+9st−8s−5t+1x9y9|x=y=1,HM1G5LSGt,s=16t+6x4y4+16t+6x4y4+32t+2x4y5+50t+2x4y5+50s−2x5y5+50s−2x5y5+502s+t−2x5y8+1282s+t−2x5y8+128sx8y8+128sx8y8+256s+t−2x8y9+324s+t−2x8y9+819st−8s−5t+1x9y9+819st−8s−5t+1x9y9+216t+6x4y4+40t+2x4y5+50s−2x5y5+802s+t−2x5y8+128sx8y8+288s+t−2x8y9+819st−8s−5t+1x9y9|x=y=1,HM1G5LSGt,s=2916st−48s+100t−2356.(12)
Fifth hyperM2 Zagreb index is defined as(140)HM2G5G=DxDyDxDyNMG|x=y=1.Now, by using equation (117), we have(141)HM2G5LSGt,s=DxDyDxDyfx,y|x=y=1,HM2G5LSGt,s=DxDyDxDyt+6x4y4+2t+2x4y5+2s−2x5y5+22s+t−2x5y8+2sx8y8+4s+t−2x8y9+9st−8s−5t+1x9y9|x=y=1,HM2G5LSGt,s=DxDy16t+6x4y4+40t+2x4y5+50s−2x5y5+802s+t−2x5y8+128sx8y8+288s+t−2x8y9+819st−8s−5t+1x9y9|x=y=1,HM2G5LSGt,s=256t+6x4y4+800t+2x4y5+1250s−2x5y5+32002s+t−2x5y8+8192sx8y8+20736s+t−2x8y9+65619st−8s−5t+1x9y9|x=y=1,HM2G5LSGt,s=59049st−15910s−7813t−40675.(13)
Fifth arithmetic-geometric index is defined as(142)AG5G=12Sx1/2Sy1/2Dx+DyNMG|x=y=1.Now, by using equation (117), we have(143)AG5LSGt,s=12Sx1/2S\y1/2Dx+Dyfx,y|x=y=1,AG5LSGt,s=12Sx1/2Sy1/2Dx+Dyt+6x4y4+2t+2x4y5+2s−2x5y5+22s+t−2x5y8+2sx8y8+4s+t−2x8y9+9st−8s−5t+1x9y9|x=y=1,AG5LSGt,s=12Sx1/2Sy1/24t+6x4y4+4t+6x4y4+8t+2x4y5+10t+2x4y5+10s−2x5y5+10s−2x5y5+102s+t−2x5y8+162s+t−2x5y8+16sx8y8+16sx8y8+32s+t−2x8y9+36s+t−2x8y9+99st−8s−5t+1x9y9+99st−8s−5t+1x9y9|x=y=1,AG5LSGt,s=8216t+6x4y4+18220t+2x4y5+20225s−2x5y5+262402s+t−2x5y8+32264sx8y8+68272s+t−2x8y9+182819st−8s−5t+1x9y9|x=y=1,AG5LSGt,s=9st−8s−5t+1+1310202s+t−2+1726s+t−2+2s−2+t+6+9510t+2+2s.(14)
Fifth geometric-arithmetic index is defined as(144)GA5G=2SxJDx1/2Dx1/2NMG|x=1.Now, by using equation (117), we have(145)GA5LSGt,s=2SxJDx1/2Dy1/2fx,y|x=1,GA5LSGt,s=2SxJDx1/2Dy1/2t+6x4y4+2t+2x4y5+2s−2x5y5+22s+t−2x5y8+2sx8y8+4s+t−2x8y9+9st−8s−5t+1x9y9|x=1,GA5LSGt,s=2168t+6x8+4209t+2x9+42510s−2x10+440132s+t−2x13+46416sx16+87217s+t−2x17+281189st−8s−5t+1x18|x=1,GA5LSGt,s=9st−8s−5t+1+810132s+t−248217s+t−2+2s−2+t+6859t+2+2s.Corollary 5.
LetLSGt,sbe the line graph of the subdivision graph of graphene network, witht≠1ands=1; then, its neighborhood degree-based graphical indices are given by(1)
M1′=192t−98(2)
M2∗=748t−599(3)
FN∗=1520t−1202(4)
M2nm=1/814t−7+1/16t+8+1/18t−1+3/20t+1/32(5)
NRα=16αt+8+20α2t+40α2t+64α2+72α4t−1+81α4t−7(6)
ND3=12256t−12030(7)
ND5=24t−7+2t+8+145/18t−1+171/20t+4(8)
NH=1/94t−7+1/4t+8+8/17t−1+88/117t+1/4(9)
NI=9/24t−7+2t+8+288/17t−1+1240/117t+8(10)
S=129.74633794t−7+512/27t+8+442.368t−1+142.8155248t+191.0670554(11)
HM1G5LSGt,s=3016t−2400(12)
HM2G5LSGt,s=51236t−56423(13)
AG5LSGt,s=4t−7+t+8+172/6t−1+4.067941659t+2(14)
GA5LSGt,s=4t−7+t+8+482/17t−1+3.93363300t+2Proof.
NM -Polynomial of LSGt,s, for t≠1 and s=1, is given by(146)NMLSGt,s=t+8x4y4+2tx4y5+2tx5y8+2x8y8+4t−1x8y9+4t−7x9y9.
Let(147)NMLSGt,s=fx,y,and then, we have(148)fx,y=NMLSGt,s=t+8x4y4+2tx4y5+2tx5y8+2x8y8+4t−1x8y9+4t−7x9y9.(1)
M1′ is defined as(149)M1′=Dx+DyNMG|x=y=1.Now, by using equation (148), we have(150)M1′=Dx+Dyfx,y|x=y=1,M1′=Dx+Dyt+8x4y4+2tx4y5+2tx5y8+2x8y8+4t−1x8y9+4t−7x9y9|x=y=1,M1′=4t+8x4y4+4t+8x4y4+8tx4y5+10tx4y5+10tx5y8+16tx5y8+16x8y8+16x8y8+32t−1x8y9+36t−1x8y9+94t−7x9y9+94t−7x9y9|x=y=1,M1′=192t−98.(2)
M2∗ is defined as(151)M2∗=DxDyNMG|x=y=1.Now, by using equation (148), we have(152)M2∗=DxDyfx,y|x=y=1,M2∗=DxDyt+8x4y4+2tx4y5+2tx5y8+2x8y8+4t−1x8y9+4t−7x9y9|x=y=1,M2∗=16t+8x4y4+40tx4y5+80tx5y8+128x8y8+288t−1x8y9+814t−7x9y9|x=y=1,M2∗=748t−599.(3)
FN∗ is defined as(153)FN∗=Dx2+Dy2NMG|x=y=1.Now, by using equation (148), we have(154)FN∗=Dx2+Dy2fx,y|x=y=1,FN∗=Dx2+Dy2t+8x4y4+2tx4y5+2tx5y8+2x8y8+4t−1x8y9+4t−7x9y9|x=y=1,FN∗=16t+8x4y4+16t+8x4y4+32tx4y5+50tx4y5+50tx5y8+128tx5y8+128x8y8+128x8y8+256t−1x8y9+324t−1x8y9+814t−7x9y9+814t−7x9y9|x=y=1,FN∗=1520t−1202.(4)
M2nm is defined as(155)M2nm=SxSyNMG|x=y=1.Now, by using equation (148), we have(156)M2nm=SxSyfx,y|x=y=1,M2nm=SxSyt+8x4y4+2tx4y5+2tx5y8+2x8y8+4t−1x8y9+4t−7x9y9|x=y=1,M2nm=116t+8x4y4+220tx4y5+240tx5y8+264x8y8+472t−1x8y9+1814t−7x9y9|x=y=1,M2nm=1814t−7+116t+8+118t−1+320t+132.(5)
NRα is defined as(157)NRα=DxαDyαNMG|x=y=1.Now, by using equation (148), we have(158)NRα=DxαDyαfx,y|x=y=1,NRα=DxαDyαt+8x4y4+2tx4y5+2tx5y8+2x8y8+4t−1x8y9+4t−7x9y9|x=y=1,NRα=16αt+8x4y4+20α2tx4y5+40α2tx5y8+64α2x8y8+72α4t−1x8y9+81α4t−7x9y9|x=y=1,NRα=16αt+8+20α2t+40α2t+64α2+72α4t−1+81α4t−7.(6)
ND3 is defined as(159)ND3=DxDyDx+DyNMG|x=y=1.Now, by using equation (148), we have(160)ND3=DxDyDx+Dyfx,y|x=y=1,ND3=DxDyDx+Dyt+8x4y4+2tx4y5+2tx5y8+2x8y8+4t−1x8y9+4t−7x9y9|x=y=1,ND3=DxDy4t+8x4y4+4t+8x4y4+8tx4y5+10tx4y5+10tx5y8+16tx5y8+16x8y8+16x8y8+32t−1x8y9+36t−1x8y9+94t−7x9y9+94t−7x9y9|x=y=1,ND3=128t+8x4y4+360tx4y5+1040tx5y8+2048x8y8+4896t−1x8y9+14584t−7x9y9|x=y=1,ND3=12256t−12030.(7)
ND5 is defined as(161)ND5=DxSy+SxDyNMG|x=y=1.Now, by using equation (148), we have(162)ND5=DxSy+SxDyfx,y|x=y=1,ND5=DxSy+SxDyt+8x4y4+2tx4y5+2tx5y8+2x8y8+4t−1x8y9+4t−7x9y9|x=y=1,ND5=t+8x4y4+t+8x4y4+85tx4y5+104tx4y5+108tx5y8+165tx5y8+2x8y8+2x8y8+329t−1x8y9+368t−1x8y9+4t−7x9y9+4t−7x9y9|x=y=1,ND5=24t−7+2t+8+14518t−1+17120t+4.(8)
NH is defined as(163)NH=2SxJNMG|x=y=1.Now, by using equation (148), we have(164)NH=2SxJfx,y|x=y=1,NH=2SxJt+8x4y4+2tx4y5+2tx5y8+2x8y8+4t−1x8y9+4t−7x9y9|x=y=1,NH=28t+8x8+49tx9+413tx13+416x16+817t−1x17+2184t−7x18|x=1,NH=194t−7+14t+8+817t−1+88117t+14.(9)
NI is defined as(165)NI=SxJDxDyNMG|x=y=1.Now, by using equation (148), we have(166)NI=SxJDxDyfx,y|x=y=1,NI=SxJDxDyt+8x4y4+2tx4y5+2tx5y8+2x8y8+4t−1x8y9+4t−7x9y9|x=y=1,NI=SxJ16t+8x4y4+40tx4y5+80tx5y8+128x8y8+288t−1x8y9+814t−7x9y9|x=y=1,NI=168t+8x8+409tx9+8013tx13+12816x16+28817t−1x17+81184t−7x18|x=1,NI=924t−7+2t+8+28817t−1+1240117t+8.(10)
S is defined as(167)S=Sx3Q−2JDx3Dy3NMG|x=y=1.Now, by using equation (148), we have(168)S=Sx3Q−2JDx3Dy3fx,y|x=y=1,S=Sx3Q−2JDx3Dy3t+8x4y4+2tx4y5+2tx5y8+2x8y8+4t−1x8y9+4t−7x9y9|x=y=1,S=4096216t+8x6+16000343tx7+1280001331tx11+5242882744x14+14929923375t−1x15+53144140964t−7x16|x=1,S=129.74633794t−7+51227t+8+442.368t−1+142.8155248t+191.0670554.(11)
Fifth hyperM1 Zagreb index is defined as(169)HM1G5G=Dx2+Dy2+2DxDyNMG|x=y=1.Now, by using equation (148), we have(170)HM1G5LSGt,s=Dx2+Dy2+2DxDyfx,y|x=y=1,HM1G5LSGt,s=Dx2+Dy2+2DxDyt+8x4y4+2tx4y5+2tx5y8+2x8y8+4t−1x8y9+4t−7x9y9|x=y=1,HM1G5LSGt,s=16t+8x4y4+16t+8x4y4+32tx4y5+50tx4y5+50tx5y8+128tx5y8+128x8y8+128x8y8+256t−1x8y9+324t−1x8y9+814t−7x9y9+814t−7x9y9+216t+8x4y4+40tx4y5+80tx5y8+128x8y8+288t−1x8y9+814t−7x9y9|x=y=1,HM1G5LSGt,s=3016t−2400.(12)
Fifth hyperM2 Zagreb index is defined as(171)HM2G5G=DxDyDxDyNMG|x=y=1.Now, by using equation (148), we have(172)HM2G5LSGt,s=DxDyDxDyfx,y|x=y=1,HM2G5LSGt,s=DxDyDxDyt+8x4y4+2tx4y5+2tx5y8+2x8y8+4t−1x8y9+4t−7x9y9|x=y=1,HM2G5LSGt,s=DxDy16t+8x4y4+40tx4y5+80tx5y8+128x8y8+288t−1x8y9+814t−7x9y9|x=y=1,HM2G5LSGt,s=256t+8x4y4+800tx4y5+3200tx5y8+8192x8y8+20736t−1x8y9+65614t−7x9y9|x=y=1,HM2G5LSGt,s=51236t−56423.(13)
Fifth arithmetic-geometric index is defined as(173)AG5G=12Sx1/2Sy1/2Dx+DyNMG|x=y=1.Now, by using equation (148), we have(174)AG5LSGt,s=12Sx1/2Sy1/2Dx+Dyfx,y|x=y=1,AG5LSGt,s=12Sx1/2Sy1/2Dx+Dyt+8x4y4+2tx4y5+2tx5y8+2x8y8+4t−1x8y9+4t−7x9y9|x=y=1,AG5LSGt,s=12Sx1/2Sy1/24t+8x4y4+4t+8x4y4+8tx4y5+10tx4y5+10tx5y8+16tx5y8+16x8y8+16x8y8+32t−1x8y9+36t−1x8y9+94t−7x9y9+94t−7x9y9|x=y=1,AG5LSGt,s=8216t+8x4y4+18220tx4y5+26240tx5y8+32264x8y8+68272t−1x8y9+182814t−7x9y9|x=y=1,AG5LSGt,s=4t−7+t+8+1726t−1+4.067941659t+2.(14)
Fifth geometric-arithmetic index is defined as(175)GA5G=2SxJDx1/2Dy1/2NMG|x=1.Now, by using Equation (16), we have(176)GA5LSGt,s=2SxJDx1/2Dy1/2fx,y|x=1,GA5LSGt,s=2SxJDx1/2Dy1/2t+8x4y4+2tx4y5+2tx5y8+2x8y8+4t−1x8y9+4t−7x9y9|x=1,GA5LSGt,s=2168t+8x8+4209tx9+44013tx13+46416x16+87217t−1x17+281184t−7x18|x=1,GA5LSGt,s=4t−7+t+8+48217t−1+3.93363300t+2.Corollary 6.
LetLSGt,sbe the line graph of the subdivision graph of graphene network, witht=1ands>1; then, its neighborhood degree-based graphical indices are given by(1)
M1′=190s−94(2)
M2∗=704s−517(3)
FN∗=1454s−1070(4)
M2nm=2/25s−2+2581/12960s−1+33/40(5)
NRα=16α10+20α4+25α2s−2+40α4s−1+64α2s−1+72α4s−1+81αs−1.(6)
ND3=10982s−9482(7)
ND5=4s−2+1033/45s−1+141/5(8)
NH=2/5s−2+1.447083962s−1+61/18(9)
NI=5s−2+18453/442s−1+260/9(10)
S=15625/256s−2+955.5179823s−1+282.9240903(11)
HM1G5LSGt,s=2862s−2104(12)
HM2G5LSGt,s=43139s−40229(13)
AG5LSGt,s=2s−2+11.11789938s−1+14.02492236(14)
GA5LSGt,s=2s−2+10.88510763s−1+13.97523196Proof.
NM-Polynomial of LSGt,s, for t=1 and s>1, is given by(177)NMLSGt,s=10x4y4+4x4y5+2s−2x5y5+4s−1x5y8+2s−1x8y8+4s−1x8y9+s−1x9y9.
Let(178)NMLSGt,s=fx,y,and then, we have(179)fx,y=NMLSGt,s=10x4y4+4x4y5+2s−2x5y5+4s−1x5y8+2s−1x8y8+4s−1x8y9+s−1x9y9.(1)
M1′ is defined as(180)M1′=Dx+DyNMG|x=y=1.Now, by using equation (179), we have(181)M1′=Dx+Dyfx,y|x=y=1,M1′=Dx+Dy10x4y4+4x4y5+2s−2x5y5+4s−1x5y8+2s−1x8y8+4s−1x8y9+s−1x9y9|x=y=1,M1′=40x4y4+40x4y4+16x4y5+20x4y5+10s−2x5y5+10s−2x5y5+20s−1x5y8+32s−1x5y8+16s−1x8y8+16s−1x8y8+32s−1x8y9+36s−1x8y9+9s−1x9y9+9s−1x9y9|x=y=1,M1′=190s−94.(2)
M2∗ is defined as(182)M2∗=DxDyNMG|x=y=1.Now, by using equation (179), we have(183)M2∗=DxDyfx,y|x=y=1,M2∗=DxDy10x4y4+4x4y5+2s−2x5y5+4s−1x5y8+2s−1x8y8+4s−1x8y9+s−1x9y9|x=y=1,M2∗=160x4y4+80x4y5+50s−2x5y5+160s−1x5y8+128s−1x8y8+288s−1x8y9+81s−1x9y9|x=y=1,M2∗=704s−517.(3)
FN∗ is defined as(184)FN∗=Dx2+Dy2NMG|x=y=1.Now, by using equation (179), we have(185)FN∗=Dx2+Dy2fx,y|x=y=1,FN∗=Dx2+Dy210x4y4+4x4y5+2s−2x5y5+4s−1x5y8+2s−1x8y8+4s−1x8y9+s−1x9y9|x=y=1,FN∗=160x4y4+160x4y4+64x4y5+100x4y5+50s−2x5y5+50s−2x5y5+100s−1x5y8+256s−1x5y8+128s−1x8y8+128s−1x8y8+256s−1x8y9+324s−1x8y9+81s−1x9y9+81s−1x9y9|x=y=1,FN∗=1454s−1070.(4)
M2nm is defined as(186)M2nm=SxSyNMG|x=y=1.Now, by using equation (179), we have(187)M2nm=SxSyfx,y|x=y=1,M2nm=SxSy10x4y4+4x4y5+2s−2x5y5+4s−1x5y8+2s−1x8y8+4s−1x8y9+s−1x9y9|x=y=1,M2nm=1016x4y4+420x4y5+225s−2x5y5+440s−1x5y8+264s−1x8y8+472s−1x8y9+181s−1x9y9|x=y=1,M2nm=225s−2+258112960s−1+3340.(5)
NRα is defined as(188)NRα=DxαDyαNMG|x=y=1.Now, by using equation (179), we have(189)NRα=DxαDyαfx,y|x=y=1,NRα=DxαDyα10x4y4+4x4y5+2s−2x5y5+4s−1x5y8+2s−1x8y8+4s−1x8y9+s−1x9y9|x=y=1,NRα=16α10x4y4+20α4x4y5+25α2s−2x5y5+40α4s−1x5y8+64α2s−1x8y8+72α4s−1x8y9+81αs−1x9y9|x=y=1,NRα=16α10+20α4+25α2s−2+40α4s−1+64α2s−1+72α4s−1+81αs−1.(6)
ND3 is defined as(190)ND3=DxDyDx+DyNMG|x=y=1.Now, by using equation (179), we have(191)ND3=DxDyDx+Dyfx,y|x=y=1,ND3=DxDyDx+Dy10x4y4+4x4y5+2s−2x5y5+4s−1x5y8+2s−1x8y8+4s−1x8y9+s−1x9y9|x=y=1,ND3=DxDy40x4y4+40x4y4+16x4y5+20x4y5+10s−2x5y5+10s−2x5y5+20s−1x5y8+32s−1x5y8+16s−1x8y8+16s−1x8y8+32s−1x8y9+36s−1x8y9+9s−1x9y9+9s−1x9y9|x=y=1,ND3=1280x4y4+720x4y5+500s−2x5y5+2080s−1x5y8+2048s−1x8y8+4896s−1x8y9+1458s−1x9y9|x=y=1,ND3=10982s−9482.(7)
ND5 is defined as(192)ND5=DxSy+SxDyNMG|x=y=1.Now, by using equation (179), we have(193)ND5=DxSy+SxDyfx,y|x=y=1,ND5=DxSy+SxDy10x4y4+4x4y5+2s−2x5y5+4s−1x5y8+2s−1x8y8+4s−1x8y9+s−1x9y9|x=y=1,ND5=10x4y4+10x4y4+165x4y5+204x4y5+2s−2x5y5+2s−2x5y5+208s−1x5y8+325s−1x5y8+2s−1x8y8+2s−1x8y8+329s−1x8y9+368s−1x8y9+s−1x9y9+s−1x9y9|x=y=1,ND5=4s−2+103345s−1+1415.(8)
NH is defined as(194)NH=2SxJNMG|x=y=1.Now, by using equation (179), we have(195)NH=2SxJfx,y|x=y=1,NH=2SxJ10x4y4+4x4y5+2s−2x5y5+4s−1x5y8+2s−1x8y8+4s−1x8y9+s−1x9y9|x=y=1,NH=208x8+89x9+410s−2x10+813s−1x13+416s−1x16+817s−1x17+218s−1x18|x=1,NH=25s−2+1.447083962s−1+6118.(9)
NI is defined as(196)NI=SxJDxDyNMG|x=y=1.Now, by using equation (179), we have(197)NI=SxJDxDyfx,y|x=y=1,NI=SxJDxDy10x4y4+4x4y5+2s−2x5y5+4s−1x5y8+2s−1x8y8+4s−1x8y9+s−1x9y9|x=y=1,NI=SxJ160x4y4+80x4y5+50s−2x5y5+160s−1x5y8+128s−1x8y8+288s−1x8y9+81s−1x9y9|x=y=1,NI=1608x8+809x9+5010s−2x10+16013s−1x13+12816s−1x16+28817s−1x17+8118s−1x18|x=1,NI=5s−2+18453442s−1+2609.(10)
S is defined as(198)S=Sx3Q−2JDx3Dy3NMG|x=y=1.Now, by using equation (179), we have(199)S=Sx3Q−2JDx3Dy3fx,y|x=y=1,S=Sx3Q−2JDx3Dy310x4y4+4x4y5+2s−2x5y5+4s−1x5y8+2s−1x8y8+4s−1x8y9+s−1x9y9|x=y=1,S=40960216x6+32000343x7+31250512s−2x8+2560001331s−1x11+5242882744s−1x14+14929923375s−1x15+5314414096s−1x16|x=1,S=15625256s−2+955.5179823s−1+282.9240903.(11)
Fifth hyperM1 Zagreb index is defined as(200)HM1G5G=Dx2+Dy2+2DxDyNMG|x=y=1.Now, by using equation (179), we have(201)HM1G5LSGt,s=Dx2+Dy2+2DxDyfx,y|x=y=1,HM1G5LSGt,s=Dx2+Dy2+2DxDy10x4y4+4x4y5+2s−2x5y5+4s−1x5y8+2s−1x8y8+4s−1x8y9+s−1x9y9|x=y=1,HM1G5LSGt,s=160x4y4+160x4y4+64x4y5+100x4y5+50s−2x5y5+50s−2x5y5+100s−1x5y8+256s−1x5y8+128s−1x8y8+128s−1x8y8+256s−1x8y9+324s−1x8y9+81s−1x9y9+81s−1x9y9+2160x4y4+80x4y5+50s−2x5y5+160s−1x5y8+128s−1x8y8+288s−1x8y9+81s−1x9y9|x=y=1,HM1G5LSGt,s=2862s−2104.(12)
Fifth hyperM2 Zagreb index is defined as(202)HM2G5G=DxDyDxDyNMG|x=y=1.Now, by using equation (179), we have(203)HM2G5LSGt,s=DxDyDxDyfx,y|x=y=1,HM2G5LSGt,s=DxDyDxDy10x4y4+4x4y5+2s−2x5y5+4s−1x5y8+2s−1x8y8+4s−1x8y9+s−1x9y9|x=y=1,HM2G5LSGt,s=DxDy160x4y4+80x4y5+50s−2x5y5+160s−1x5y8+128s−1x8y8+288s−1x8y9+81s−1x9y9|x=y=1,HM2G5LSGt,s=2560x4y4+1600x4y5+1250s−2x5y5+6400s−1x5y8+8192s−1x8y8+20736s−1x8y9+6561s−1x9y9|x=y=1,HM2G5LSGt,s=43139s−40229.(13)
Fifth arithmetic-geometric index is defined as(204)AG5G=12Sx1/2Sy1/2Dx+DyNMG|x=y=1.Now, by using equation (179), we have(205)AG5LSGt,s=12Sx1/2Sy1/2Dx+Dyfx,y|x=y=1,AG5LSGt,s=12Sx1/2Sy1/2Dx+Dy10x4y4+4x4y5+2s−2x5y5+4s−1x5y8+2s−1x8y8+4s−1x8y9+s−1x9y9|x=y=1,AG5LSGt,s=12Sx1/2Sy1/240x4y4+40x4y4+16x4y5+20x4y5+10s−2x5y5+10s−2x5y5+20s−1x5y8+32s−1x5y8+16s−1x8y8+16s−1x8y8+32s−1x8y9+36s−1x8y9+9s−1x9y9+9s−1x9y9|x=y=1,AG5LSGt,s=80216x4y4+36220x4y5+20225s−2x5y5+52240s−1x5y8+32264s−1x8y8+6872s−1x8y9+18281s−1x9y9|x=y=1,AG5LSGt,s=2s−2+11.11789938s−1+14.02492236.(14)
Fifth geometric-arithmetic index is defined as(206)GA5G=2SxJDx1/2Dy1/2NMG|x=1.Now, by using equation (179), we have(207)GA5LSGt,s=2SxJDx1/2Dy1/2fx,y|x=1,GA5LSGt,s=2SxJDx1/2Dy1/210x4y4+4x4y5+2s−2x5y5+4s−1x5y8+2s−1x8y8+4s−1x8y9+s−1x9y9|x=1,GA5LSGt,s=20168x8+8209x9+42510s−2x10+84013s−1x13+46416s−1x16+87217s−1x17+28118s−1x18|x=1,GA5LSGt,s=2s−2+10.88510763s−1+13.97523196.


## 5. Conclusions

Graphene is an allotrope of carbon consisting of a single layer of atoms arranged in a two-dimensional honeycomb lattice nanostructure. It was an interesting problem to give an easy way to compute its topological indices. In this article, we established NM-polynomials for the graph of graphene and the line graph of the subdivision graph of graphene. By applying some basic roles of calculus, we computed serval of its neighborhood-based indices. The established NM-polynomials can also be used to compute the remaining neighborhood-based topological indices.



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*Source: 3809806-2022-11-12.xml*