On Molecular Descriptors of Carbon Nanocones - MDPI

biomolecules Article

On Molecular Descriptors of Carbon Nanocones Waqas Nazeer 1, * , Adeel Farooq 2 , Muhammad Younas 2 , Mobeen Munir 1 Shin Min Kang 3,4, * 1 2 3 4

*

and

Division of Science and Technology, University of Education, Lahore 54000, Pakistan; [email protected] Lahore Campus, COMSATS University Islamabad, Lahore 54000, Pakistan; [email protected] (A.F.); [email protected] (M.Y.) Department of Mathematics and RINS, Gyeongsang National University, Jinju 52828, Korea Center for General Education, China Medical University, Taichung 40402, Taiwan Correspondence: [email protected] (W.N.); [email protected] (S.M.K.)

Received: 14 August 2018; Accepted: 31 August 2018; Published: 7 September 2018

Abstract: Many degree-based topological indices can be obtained from the closed-off M-polynomial of a carbon nanocone. These topological indices are numerical parameters that are associated with a structure and, in combination, determine the properties of the carbon nanocone. In this paper, we compute the closed form of the M-polynomial of generalized carbon nanocone and recover many important degree-based topological indices. We use software Maple 2015 (Maplesoft, Waterloo, ON, Canada) to plot the surfaces and graphs associated with these nanocones, and relate the topological indices to the structure of these nanocones. Keywords: M-polynomial; degree-based topological index; carbon nanocone

1. Introduction Chemical graph theory is a subject that connects mathematics, chemistry, and graph theory, and solves problems arising in chemistry mathematically. A topological index is a numeric number associated with a molecular graph that correlates certain physicochemical properties of chemical compounds. The topological indices are useful in the prediction of physicochemical properties and the bioactivity of the chemical compounds [1–4]. These indices capture the overall structure of the compound and predict chemical properties such as strain energy, heat formation, boiling points, etc. Long computation is required to compute topological indices, and in order to simplify the computation of the degree indices, which form a subclass of degree-based topological indices of utmost importance, the M-polynomial was introduced in [5] by Deutsch and Klavžar. From the M-polynomial, one can recover nine degree-based topological indices. For this reason, the M-polynomial has been studied extensively in recent years, for example, Munir et al. computed M-polynomials for polyhex nanotubes, and recovered many important degree-based topological indices [6]. The M-polynomials for nanostar dendrimers were studied by Munir et al. in [7]. Munir et al. also studied the M-polynomials of titania nanotubes and its degree-based topological indices [8]. Ali et al. [9] studied zigzag and rhombic benzenoid systems. For more studies in this direction, see Munir et al. [10], Kwun et al. [11], Kang et al. [12], Ahmad et al. [13] and Kang et al. [14]. Carbon nanocones have been observed since 1968 or even earlier [15] on the surface of naturally occurring graphite. The molecular graph of nanocones have conical structures with a cycle of length k at its core and n layers of hexagons placed at the conical surface around its center. The importance of carbon nanostructures is due to their potential use in many applications, including gas sensors, energy storage, nanoelectronic devices, biosensors, and chemical probes [16]. Carbon allotropes such as carbon nanocones and carbon nanotubes have been proposed as possible molecular gas storage

Biomolecules 2018, 8, 92; doi:10.3390/biom8030092

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devices [17]. More recently, carbon nanocones have gained increased scientific interest due to their unique unique properties and promising uses in many novel applications such as energy unique properties properties and and promising promising uses uses in in many many novel novel applications applications such such as as energy energy and and hydrogen hydrogen storage. Figures 1 and 2 show carbon nanocones. storage. Figures Figures 1 and 2 show carbon nanocones.

Figure for Figure1. Carbonnanocone nanoconeCNC CNCkkk [nnn]for Figure 1.1.Carbon Carbon nanocone fork kk===5.5. 5.

Figure 2.The The moleculargraph graph ofCNC CNC [n] for k k==5.5. Figure Figure 2. 2. The molecular molecular graph of of CNCkkk nn for for k = 5.

The molecular graph of CNCk [n] nanocones have conical structures with a cycle of length k at its CNC The molecular graph nanocones have aa cycle kk at CNCkk nat n the The n molecular graph of of placed nanocones have conical conical structures with cycle of length length at core and layers of hexagons conical surface aroundstructures its center, with as shown in of the following its core and n layers of hexagons placed at the conical surface around its center, as shown in the Figure 3. its core and n layers of hexagons placed at the conical surface around its center, as shown in the following 3. In theFigure present following Figure 3.report, we give a closed form of the M-polynomial of carbon nanocones. From the In the present we give aa closed of carbon From M-polynomial, we recover nine indices. Inof Xu nanocones. et al. computed the In the present report, report, we givedegree-based closed form formtopological of the the M-polynomial M-polynomial of[18], carbon nanocones. From the the M-polynomial, we recover nine degree-based topological indices. In [18], Xu et al. computed the Hosoya polynomial and related distance-based indices for CNC n . In [19], Ghorbani et al. computed [ ] 7 M-polynomial, we recover nine degree-based topological indices. In [18], Xu et al. computed the Hosoya polynomial and related distance-based for 𝐶𝑁𝐶 In Ghorbani et computed the Vertex PI, Szeged, and Omega polynomials indices of carbon many partial ]. Similarly, [𝑛].CNC 4 [ nGhorbani Hosoya polynomial and related distance-based indices fornanocone 𝐶𝑁𝐶77[𝑛]. In [19], [19], et al. al. computed [𝑛]. the Vertex PI, Szeged, and Omega polynomials of carbon nanocone 𝐶𝑁𝐶 Similarly, many partial results regarding topological indices have been obtained for some particular classes of nanocones. 4 the Vertex PI, Szeged, and Omega polynomials of carbon nanocone 𝐶𝑁𝐶4 [𝑛]. Similarly, many partial results topological indices have been classes of However, we present some general results complete families nanocones. Our results present results regarding regarding topological indices haveabout been obtained obtained for for some someofparticular particular classes of nanocones. nanocones. However, we present some general results about complete families of nanocones. Our results present organized generalizations many existing partialcomplete results. families of nanocones. Our results present However, we present someofgeneral results about organized generalizations of many existing partial organized generalizations of many existing partial results. results.

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Figure 3. Carbon nanocone CNC [n]. Figure 3. Carbon nanocone CNCkk  n .

.

2.2.Basic BasicDefinitions Definitionsand andNotions Notions Algebraic Algebraic polynomials polynomials have have many many useful useful applications applications in inchemistry. chemistry. For For instance, instance, the theHosoya Hosoya polynomial polynomial(also (alsocalled calledthe theWiener Wienerpolynomial) polynomial)[20] [20]plays playsaavital vitalrole rolein indetermining determiningdistance-based distance-based topological topological indices. indices. The The M-polynomial M-polynomial [5], [5], which which was was introduced introduced in in 2015, 2015, plays plays the the same same role role in in determining many degree-based topological indices [6–14]. determining many degree-based topological indices [6–14]. Throughout Throughoutthis thispaper, paper,GGdenotes denotesconnected connectedgraph, graph,V(G) V(G)and andE(G) E(G)denote denotethe thevertex vertexset setand andthe the edge set, respectively, and d denotes the degree of a vertex. v edge set, respectively, and d denotes the degree of a vertex. v

Definition Definition1.1.The TheM-polynomial M-polynomialofofGGisisdefined definedas: as: MM( G,  Gx,, xy, )y =

∑ 

δ≤ii≤ ∆  jj≤ 

mijij (G G) xxii yy jj, , m

where , ∆ =Max G)} and mijand ( G ) is E( G vu ) such {dv |dvv ∈| vV(G {dv |vd∈v V  Ethat (G) whereδ = the∈edge mijthe (Gedge ) is vu  Min Min{ V)}(G)},  Max{ | v( V, (G)}, d , d = i, j . { v u} { } such that dv , du   i, j . The first well-known topological index was introduced by Wiener [21] when he was studying The first well-known topological index waspath introduced Wiener [21] when heaswas the boiling point of paraffin. He named it the number,by which is now known thestudying Wiener the boiling point of paraffin. He named it the path number, which is now known as the Wiener index index [22,23]. Later, Randic defined the first degree-based topological index in 1975 [24]. The Randic [22,23]. Later, Randic degree-based topological index in 1975 [24]. The Randic index index is denoted by R−defined andfirst is defined as: 1/2 ( G ), the is denoted by R1/2 (G) , and is defined as: 1 R−1/2 ( G ) = ∑ √ 1 . d u dv. R1/2 (G)  uv∈ E( G ) d u dv uvE (G ) In 1998, working independently, Bollobas and Erdos [25] and Amic et al. [26] proposed In 1998, working independently, and studied Erdos [25] and Amicby et al. [26]chemists proposedand the the generalized Randic index, whichBollobas has been extensively both generalized Randic has been properties studied extensively by both chemists and mathematicians mathematicians [27].index, Manywhich mathematical of the Randic index have been discussed [28]. [27]. Many mathematical properties of the Randic index have been discussed [28]. For a detailed For a detailed survey we refer to the monograph of Li and Gutman [29]. survey we refer to the monograph of Li and Gutman [29]. The general Randic index is defined as:


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The general Randic index is defined as: Rα ( G ) =

∑

(du dv )α .

uv∈ E( G )

Obviously R−1/2 ( G ) is the particular case of Rα ( G ) when α = − 12 . Gutman and Trinajstic introduced the first Zagreb index and second Zagreb index, which are defined as: M1 ( G ) = ∑ (du + dv ) and M2 ( G ) = ∑ (du × dv ), respectively. The second uv∈ E( G )

uv∈ E( G )

modified Zagreb index was defined as: m

M2 ( G ) =

∑

uv∈ E( G )

1 . d(u)d(v)

For detail about these indices, we refer Nikolić et al. [30], Gutman and Das [31], Das and Gutman [32] and Trinajstić et al. [33] to the readers. The symmetric division index is defined as: SDD ( G ) =

∑

uv∈ E( G )

min(du , dv ) max(du , dv ) + . max(du , dv ) min(du , dv )

Other well-known topological indices are the harmonic index, H ( G ) =

∑

vu∈ E( G )

sum index, I ( G ) =

∑

vu∈ E( G )

du dv du +dv ,

2 du +dv .,

the inverse

and the augmented Zagreb index [34,35]:

A( G ) =

∑

vu∈ E( G )

du dv du + dv − 2

3 .

The following Table 1 relates some well-known degree-based topological indices with the M-polynomial [5]. Table 1. Derivation of some degree-based topological indices from the M-polynomial. Derivation from M( G; x, y) ( Dx + Dy )( M( G; x, y)) x=y=1 ( Dx Dy )( M( G; x, y)) x=y=1 (Sx Sy )( M( G; x, y)) x=y=1 ( Dxα Dyα )( M( G; x, y)) x = y =1 (Sαx Syα )( M( G; x, y)) x =y= 1 ( Dx Sy + Sx Dy )( M( G; x, y))

Topological Index First Zagreb Second Zagreb Second Modified Zagreb General Randić Index General inverse Randić Index Symmetric Division Index Harmonic Index Inverse sum Index Augmented Zagreb Index

x = y =1

Sx 3

2 Sx J ( M( G ; x, y)) x=1 Sx J Dx Dy ( M( G ; x, y)) x=1 Q−2 J Dx 3 Dy 3 ( M ( G ; x , y)) x=1

where: Dx = x

∂( f ( x,y) ∂x ,

Dy = y

∂( f ( x,y) ∂y ,

Sx =

Rx 0

f (t,y) t dt, Sy

Qα ( f ( x, y)) = x α f ( x, y). 3. Results In this section, we give our computational results.

=

Ry 0

f ( x,t) t dt,

J ( f ( x, y)) = f ( x, x ),


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Theorem 1. Let CNCk [n] be the graph of carbon nanocones. Then, the M-polynomial of CNCk [n] is: M(CNCk [n], x, y) = kx2 y2 + 2knx2 y3 +

kn (3n + 1) x3 y3 . 3

Proof. Let CNCk [n] be the graph of nanocones. From the graph of CNCk [n] nanocones, we can see that there are two partitions, V{3} = {v ∈ V (CNCk [n])|dv = 3} and V{2} = {v ∈ V (CNCk [n])|dv = 2}. The edge set of the CNCk [n] can be partitions as follows: E{2,2} = {e = uv ∈ E(CNCk [n])|du = 2&dv = 2}, E{2,3} = {e = uv ∈ E(CNCk [n])|du = 2&dv = 3} and: E{3,3} = {e = uv ∈ E(CNCk [n])|du = dv = 3} From the molecular graph of CNCk [n], we can observe that E{2,2} = k, E{2,3} = 2kn, and E{3,3} = kn 3 (3n + 1). Thus, by Definition 1, the M-polynomial of CNCk [n] (Figure 4) is: M (CNCk [n]; x, y)

= ∑ mij (CNCk [n]) xi y j , i≤ j

= ∑ m22 (CNCk [n]) x2 y2 + ∑ m23 (CNCk [n]) x2 y3 + ∑ m33 (CNCk [n]) x3 y3 , 2≤3

2≤2

3≤3

m22 (CNCk [n]) x2 y2 + ∑ m23 (CNCk [n]) x2 y3 + ∑ m33 (CNCk [n]) x3 y3 , uv∈ E{2,2} uv∈ E{3,3} uv∈E{2,3 } = E{2,2} x2 y2 + E{2,3} x2 y3 + E{3,3} x3 y3 ,

=

∑

= kx2 y2 + 2knx2 y3 +

kn 3 3 3 (3n + 1) x y .

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CNC Figure 4. The plotfor forthe the M-polynomial M-polynomial of of k =kn== n 1. = 1. Figure 4. The 3D3D plot CNC ] for k  nk[ nfor Theorem 2. Let CNCk  n be the graph of carbon nanocones. Then: 1.

M1  CNCk  n  6kn2  12kn  4k.

2.

M 2  CNCk  n  9kn2  15kn  4k.

3.

m

M 2  CNCk  n 

1 2 10 1 kn  kn  k. 9 27 4 2

2

2 1

2 1


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Theorem 2. Let CNCk [n] be the graph of carbon nanocones. Then: 1. 2. 3. 4. 5. 6. 7. 8. 9.

M1 (CNCk [n]) = 6kn2 + 12kn + 4k. M2 (CNCk [n]) = 9kn2 + 15kn + 4k. m M (CNC [ n ]) = 1 kn2 + 10 kn + 1 k. 2 k 9 27 4 Rα (CNCk [n]) = k22α + kn n32α + 22α+1 + 32α−1 . Rα (CNCk [n]) = k4−α + 2kn6−α + 13 kn(3n + 1)9−α . SDD (CNCk [n]) = 2kn2 + 5kn + 2k. 1 H (CNCk [n]) = 13 kn2 + 41 45 kn + 2 k. I (CNCk [n]) = 23 kn2 + 29 10 kn + k. 2 + 1267 kn + 8k. A(CNCk [n]) = 729 kn 64 64

Proof. Let M (CNCk [n]; x, y) = f ( x, y) = kx2 y2 + 2knx2 y3 +

kn 3 3 3 (3n + 1) x y ,

Dx ( f ( x, y)) = 2kx2 y2 + 4knx2 y3 + kn(3n + 1) x3 y3 , Dy ( f ( x, y)) = 2kx2 y2 + 6knx2 y3 + kn(3n + 1) x3 y3 , Sx ( f ( x, y)) = Sy ( f ( x, y)) =

kn k 2 2 x y + knx2 y3 + (3n + 1) x3 y3 , 2 9

kn k 2 2 2 x y + knx2 y3 + (3n + 1)mx3 y3 , 2 3 9

Dxα Dyα ( f ( x, y)) = 22α kx2 y2 + 2α+1 3α knx2 y3 + 32α−1 kn(3n + 1) x3 y3 , Sαx Syα ( f ( x, y)) =

kn k 2 2 1 x y + α−1 α knx2 y3 + 2α+1 (3n + 1) x3 y3 22α 2 3 3

J ( f ( x, y)) = kx4 + 2knx5 + Sx J ( f ( x, y)) =

kn (3n + 1) x6 , 3

k 4 2 kn x + knx5 + (3n + 1) x6 , 4 5 18

Dx Dy ( f ( x, y)) = 4kx2 y2 + 12knx2 y3 + 3kn(3n + 1) x3 y3 , JDx Dy ( f ( x, y)) = 4kx4 + 12knx5 + 3kn(3n + 1) x6 , Sx JDx Dy ( f ( x, y)) = kx4 +

1 12 knx5 + kn(3n + 1) x6 , 5 2

Dy3 ( f ( x, y)) = 8kx2 y2 + 54knx2 y3 + 9kn(3n + 1) x3 y3 , Dx3 Dy3 ( f ( x, y)) = 64kx2 y2 + 432knx2 y3 + 243kn(3n + 1) x3 y3 , JDx3 Dy3 ( f ( x, y)) = 64kx4 + 432knx5 + 243kn(3n + 1) x6 , Q−2 JDx3 Dy3 ( f ( x, y)) = 64kx2 + 432knx3 + 243kn(3n + 1) x4 , S3x Q−2 JDx3 Dy3 ( f ( x, y)) = 8kx2 + 16knx3 +

243 kn(3n + 1) x4 . 64

Now we recover degree-based topological indices by using Table 1. The Figures 5–13 show the relations of different topological indices with values of k and n. It is noticeable that all of the above discussed topological indices vary quadratically with n, and linearly with k. 1. M1 (CNCk [n]) = Dx + Dy ( f ( x, y))( M(CNCk [n]) x, y) x=y=1 = 2k 18n2 + 26n + 10 .

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Figure 5. Plots for the first Zagreb index (left for arbitrary n and k, middle for k = 4, and right for nFigure = 5). 5. Plots for the first Zagreb index (left for arbitrary n and k, middle for k = 4, and right for n = 5).

2. 2. 2.

3. 3.

2 middle Figure 5. 5. Plots Plots for forthe thefirst firstZagreb Zagrebindex index(left (leftfor forarbitrary arbitraryn nand and middle = 4, and right Figure forfor k =k 4, and right forfor n = 5). kk, k, 4 3 2 2 M CNC n  D D f x , y M CNC n x , y |  81 n  234 n  255 n  126 n  24 .               n =2 5). k x y k x  y 1 k 4 3 2 DyD  f(f x(,x,y y))( n 4+ 234 n 3 255 n 2126 n  24+ .24.  xx,, yy) |xx=yy=11 =4 k42 8181n  MM(CNC 2 2CNC k kn[n])=DxD k k[nn]) 2.MM CNC 234n + 255n + 126n (CNC x y 4

 



  f  x, y   M CNCk n x, y  |x y 1

M 2  CNCk  n  Dx Dy

  k  81n 4 2

4

3

 234n  255n

2

   126n  24  .

Figure 6. Plots for the second Zagreb index (left for arbitrary n and k, middle for k = 4, and right for Figure Plots for for the the second second Zagreb index (left for arbitrary Figure arbitrary nn and and k, k, middle middle for forkk==4,4,and andright rightfor for n = 5). 6.6.Plots n = 5). n = 5). (left Figure 6. Plots for the second Zagreb index n1and k, middle for k = 4, and right for 1 for2 arbitrary 10 m 1 2 kn 10 m2MCNC M  = S x SSyx S yf (xf,(yx, y|x)) y 1  1kn  kn k. 1 k  nk [ n ]) + 3. 10 2 (CNC x =y=91 = 29 kn nm= 5). 27 27 41 + 4 k.



 



 f x, y |  kn     x y 1 9





M 2  CNCk  n  S x S y

m

3.

 

M 2  CNCk  n  S x S y



kn  k. 27 4 1 2 10 1  f  x, y  |x y 1  9 kn  27 kn  4 k.

7. Plots Plotsfor forthe themodified modifiedsecond second Zagreb index (left arbitrary n and k, middle = 4, Figure 7. Zagreb index (left forfor arbitrary n and k, middle for kfor = 4,k and Figure Plots and right n =for 5).the modified second Zagreb index (left for arbitrary n and k, middle for k = 4, and right for7.nfor = 5). right for n = 5).

4. 4. 4.

     2 R  CNCk  n  D  kn  n32  22 1  32 1  . x Dy f  x, y  |x  y 1  k 2

α D α f ( x, y ) 2α + kn n32α + 22α+1 + 32α−1 . 4. Rα (CNC =2k2index [nfor ]) = Figure 7. Plots the second Zagreb (left  Dmodified  2 for2arbitrary  1 2n1and k, middle for k = 4, and x = y = 1 k x y R  CNCk  n  Dx Dy f  x, y  |x  y 1  k 2  kn n3  2 3 .  2 right for nk=5). R  CNC n  D  kn n32  22 1  32 1 . x Dy f  x, y  |x  y 1  k 2

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Figure 8. Plots for the Randic index (left for arbitrary n and k, middle for k = 4, and right for n = 5). Figure 8. Plots for the Randic index (left for arbitrary n and k, middle for k = 4, and right for n = 5). Figure8.8. Plots Plotsfor forthe theRandic Randicindex index(left (leftfor forarbitrary arbitrarynnand andk,k,middle middlefor forkk==4, 4,and andright rightfor fornn== 5). 5). Figure 1

5. 5. 5.





R  CNCk  n  S  S  f  x, y   |x  y 1  k 4  2kn6 1 kn(3n  1)9 . x y     3kn RR CNC − 2kn 62kn6  − . −α α+ α (3n1 1)9 +  ]) =S x SSyαx Syαf (xf,(yx,y|x)) yx=1 y=k14= k  n[ n 5. (CNC k4  31 + 3 kn (3n  1)9 . R αCNCk kn  S  S f x , y |  k 4  2 kn 6  kn (3 n  1 ) 9 .     x y x  y 1 3

Figure 9. Plots for the inverse Randic index (left for arbitrary n and k, middle for k = 4, and right for Figure for the the inverse inverse Randic Randicindex index(left (leftfor forarbitrary arbitrarynnand andk,k,middle middlefor fork k==4,4,and andright rightfor for n = 5). 9. Plots for Figure 9. Plots for the inverse Randic index (left for arbitrary n and k, middle for k = 4, and right for n == 5). 5). n = 5).

6. 6. 6.

  

  

SDD  CNCk  n  Dx S y  Dy S x  f  x, y   |x  y 1  2kn22  5kn2  2k. 6. SDD (CNC k [ n ]) = Dx Sy + Dy S x ( f ( x, y )) x =y=1 = 2kn + 5kn + 2k. SDD CNC k  n  Dx S y  Dy S x  f  x, y   |x  y 1  2kn  5kn  2k. SDD  CNCk  n  Dx S y  Dy S x  f  x, y   |x  y 1  2kn2  5kn  2k.

Figure10. 10.Plots Plotsfor forthe thesymmetric symmetricdivision divisionindex index(left (leftfor forarbitrary arbitrarynnand andk,k,middle middlefor forkk==4,4,and andright right Figure Figure 10. Plots for the symmetric division index (left for arbitrary n and k, middle for k = 4, and right fornn==5). 5). for Figure 10. Plots for the symmetric division index (left for arbitrary n and k, middle for k = 4, and right for n = 5). for n = 5). 41 1 1 2

7. 7. 7.

7. H (CNCk [n]) = 2Sx J ( f ( x, y)) x=11= 32kn 41+ 45 kn 1 + k. H  CNCk  n  2S x J  f  x, y   |x 1 1 kn 41 kn 1 k. 2 2 3 45 H  CNCk  n  2S x J  f  x, y   |x 1  1kn 2  41kn  12k. H  CNCk  n  2S x J  f  x, y   |x 1  3 kn  45 kn  2 k. 3 45 2


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Figure 11. Plots for the harmonic index (left for arbitrary n and k, middle for k = 4, and right for Figure n = 5). 11. Plots for the harmonic index (left for arbitrary n and k, middle for k = 4, and right for n = 5).

8. 8. 8.

 

 

29 nnand 2arbitrary Figure 11. harmonic index index (left (left3for for arbitrary andk,k,middle middlefor forkk==4,4,and andright rightfor forn = 5). Figure 11.Plots Plots for for the the harmonic I CNC k n  S x JDx Dy  f  x, y   |x 1  3 kn  29 kn  k . 2 2 10 nI= CNC 5).  n  S JD D  f  x, y   |  kn  kn  k. k

x

x

x 1

y

8. I (CNCk [n]) = Sx JDx Dy ( f ( x, y)) x=12 = 32 kn120+ 29 10 kn + k. 3 2 29 I  CNCk  n  S x JDx Dy  f  x, y   |x 1  kn  kn  k. 2 10

Figure 12. Plots for the inverse sum index (left for arbitrary n and k, middle for k = 4, and right for Figure 12. Plots for the forfor k =k 4, right forfor Figure the inverse inverse sum sumindex index(left (leftfor forarbitrary arbitrarynnand andk,k,middle middle = and 4, and right n = 5). 12. nn == 5).

9. 9. 9.

 

 

for 729arbitrary 2 1267 Figure 12. Plots inverse n and middle for k = 4, and right for A ACNC for S=x3the Q JDx3JD D3y3sum f 3(xindex , (yx, kn kn k, 8 k.  y|x(left 729  2  k  n[n ]) 32Q 1  729 + 1267 9. CNC S D f = ( )) 1267 − 2 x = 1 k x x y 64 64 3 3 3 2 kn 64 64 kn + 8k. nA= 5). CNC  n  S Q JD D  f  x, y   |  kn  kn  8k. k

x 2

x

y

x 1

64 64 729 1267 A  CNCk  n  S x3Q2 JDx3 D3y  f  x, y   |x 1  kn2  kn  8k. 64 64

Figure13. 13.Plots Plotsfor forthe theaugmented augmentedZagreb Zagrebindex index(left (leftfor forarbitrary arbitrarynnand andk,k,middle middlefor forkk==4,4,and andright right Figure Figure 13. Plots for the augmented Zagreb index (left for arbitrary n and k, middle for k = 4, and right fornn== 5). 5). for for n = 5).

4.4. Conclusions Conclusions Figure 13. Plots for the augmented Zagreb index (left for arbitrary n and k, middle for k = 4, and right 4. Conclusions for nclosed = 5). form of the M-polynomials of all of the carbon nanocones is computed. This polynomial The The closed form of the M-polynomials of all of the carbon nanocones is computed. This generates a lot of form information degree-based topological descriptors, which arecomputed. actually graph The closed theofabout M-polynomials of all of the carbon nanocones is This polynomial generates aoflot information about degree-based topological descriptors, which are 4. Conclusions invariants. These indices, in combination, determine the properties of nanocones. The topological polynomial generates a lot of information about degree-based topological descriptors, which The are actually graph invariants. These indices, in combination, determine the properties of nanocones. indices in this paper arepaper important guessing the the physicochemical properties of actually graph invariants. These indices, inare combination, determine properties nanocones. The Thecalculated closed form of the M-polynomials of for all of the carbon nanocones isofcomputed. This topological indices calculated in this important for guessing the physicochemical properties understudy chemical compounds. For example, the Randi´ c index is a topological descriptor that topological indices calculated this paper are important for guessing the properties polynomial generates a lot ofininformation about degree-based topological descriptors, which are of understudy chemical compounds. For example, the Randić index is aphysicochemical topological descriptor that has beengraph connected withcompounds. numerous substance qualities of atoms, and been found to be parallel of understudy chemical Forinexample, the Randić index a topological descriptor that actually invariants. These indices, combination, determine theishas properties of nanocones. The topological indices calculated in this paper are important for guessing the physicochemical properties of understudy chemical compounds. For example, the Randić index is a topological descriptor that


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to processing the boiling point and Kovats constants of the particles. To associate with certain physicochemical properties, the GA index has a very preferable prescient control over the prescient intensity of the Randić index. The first and second Zagreb indexes were found to calculate the aggregate π-electron vitality of the atoms inside specific surmised articulations. These are among the graph invariants, which were proposed for the estimation of the skeleton of the spreading of the carbon molecule. To calculate the distance-based topological indices of understudied nanocones is an interesting problem that is worthy of further investigation. Author Contributions: W.N. design the problem. A.F. and M.Y. investigate the results, M.M. draw the figures and S.M.K. formate the paper and validate the results. Funding: This research received no external funding. Acknowledgments: The authors are thankful to both the reviewers for suggestion that help us to improve this paper. Conflicts of Interest: The authors declare no conflict of interest.

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