# M-Polynomial and Degree Based Topological Indices of Some Nanostructures

^{*}

## Abstract

**:**

## 1. Introduction and Preliminaries

## 2. Materials and Methods

## 3. Results

#### 3.1. M-Polynomials of the Dendrimer Nanostars

**Theorem**

**1.**

**Proof.**

**Theorem**

**2.**

**Proof.**

#### 3.2. M-Polynomials of the Nanotubes

**Theorem**

**3.**

**Proof.**

**Theorem**

**4.**

**Proof.**

**Theorem**

**5.**

**Proof.**

**Theorem**

**6.**

**Proof.**

#### 3.3. Degree-Based Topological Indices

**Theorem**

**7.**

- 1
- ${M}_{1}(G)=204\times {2}^{n-1}-78$
- 2
- ${M}_{2}(G)=246\times {2}^{n-1}-99$
- 3
- ${}^{m}{M}_{2}(G)=\frac{23}{3}\times {2}^{n-1}-\frac{7}{3}$
- 4
- ${R}_{\alpha}(G)={4}^{\alpha}(12\times {2}^{n-1})+{6}^{\alpha}(24\times {2}^{n-1}-12)+{9}^{\alpha}(6\times {2}^{n-1}-3)$
- 5
- ${R}_{\alpha}(G)=\frac{1}{{4}^{\alpha}}(12\times {2}^{n-1})+\frac{1}{{6}^{\alpha}}(24\times {2}^{n-1}-12)+\frac{1}{{9}^{\alpha}}(6\times 26n-1-3)$
- 6
- $SDD(G)=88\times {2}^{n-1}-32$

**Proof.**

$\begin{array}{cc}\hfill {D}_{x}(g(x,y))& =(24\times {2}^{n-1}){x}^{2}{y}^{2}+(48\times {2}^{n-1}-24){x}^{2}{y}^{3}\hfill \\ & +(18\times {2}^{n-1}-9){x}^{3}{y}^{3}\hfill \\ \hfill {D}_{y}(g(x,y))& =(24\times {2}^{n-1}){x}^{2}{y}^{2}+(72\times {2}^{n-1}-36){x}^{2}{y}^{3}\hfill \\ & +(18\times {2}^{n-1}-9){x}^{3}{y}^{3}\hfill \\ \hfill {S}_{x}(g(x,y))& =(6\times {2}^{n-1}){x}^{2}{y}^{2}+(12\times {2}^{n-1}-6){x}^{2}{y}^{3}\hfill \\ & +(2\times {2}^{n-1}-1){x}^{3}{y}^{3}\hfill \\ \hfill {S}_{y}(g(x,y))& =(6\times {2}^{n-1}){x}^{2}{y}^{2}+(8\times {2}^{n-1}-4){x}^{2}{y}^{3}\hfill \\ & +(2\times {2}^{n-1}-1){x}^{3}{y}^{3}\hfill \end{array}$ |

$\begin{array}{cc}\hfill {D}_{x}{D}_{y}(g(x,y))& =(48\times {2}^{n-1}){x}^{2}{y}^{2}+(144\times {2}^{n-1}-72){x}^{2}{y}^{3}\hfill \\ & +(54\times {2}^{n-1}-27){x}^{3}{y}^{3}\hfill \\ \hfill {S}_{x}{S}_{y}(g(x,y))& =(3\times {2}^{n-1}){x}^{2}{y}^{2}+(4\times {2}^{n-1}-2){x}^{2}{y}^{3}\hfill \\ & +(\frac{2}{3}\times {2}^{n-1}-\frac{1}{3}){x}^{3}{y}^{3}\hfill \\ \hfill {D}_{x}{S}_{y}(g(x,y))& =(12\times {2}^{n-1}){x}^{2}{y}^{2}+(16\times {2}^{n-1}-8){x}^{2}{y}^{3}\hfill \\ & +(6\times {2}^{n-1}-3){x}^{2}{y}^{3}\hfill \\ \hfill {D}_{y}{S}_{x}(g(x,y))& =(12\times {2}^{n-1}){x}^{2}{y}^{2}+(36\times {2}^{n-1}-18){x}^{2}{y}^{3}\hfill \\ & +(6\times {2}^{n-1}-3){x}^{2}{y}^{3}\hfill \end{array}$ |

$\begin{array}{cc}\hfill {D}_{y}^{\alpha}(g(x,y))& ={2}^{\alpha}(12\times 2n-1){x}^{2}{y}^{2}+{3}^{\alpha}(24\times {2}^{n-1}-9){x}^{2}{y}^{3}\hfill \\ & +{3}^{\alpha}(6\times {2}^{n-1}-3){x}^{3}{y}^{3}\hfill \\ \hfill {S}_{y}^{\alpha}(g(x,y))& =\frac{1}{{2}^{\alpha}}(12\times 2n-1){x}^{2}{y}^{2}+\frac{1}{{3}^{\alpha}}(24\times {2}^{n-1}-9){x}^{2}{y}^{3}\hfill \\ & +\frac{1}{{3}^{\alpha}}(6\times {2}^{n-1}-3){x}^{3}{y}^{3}\hfill \end{array}$ |

$\begin{array}{cc}\hfill {D}_{x}^{\alpha}{D}_{y}^{\alpha}(g(x,y))& ={4}^{\alpha}(12\times 2n-1){x}^{2}{y}^{2}+{6}^{\alpha}(24\times {2}^{n-1}-9){x}^{2}{y}^{3}\hfill \\ & +{9}^{\alpha}(6\times {2}^{n-1}-3){x}^{3}{y}^{3}\hfill \\ \hfill {S}_{x}^{\alpha}{S}_{y}^{\alpha}(g(x,y))& =\frac{1}{{4}^{\alpha}}(12\times 2n-1){x}^{2}{y}^{2}+\frac{1}{{6}^{\alpha}}(24\times {2}^{n-1}-9){x}^{2}{y}^{3}\hfill \\ & +\frac{1}{{9}^{\alpha}}(6\times {2}^{n-1}-3){x}^{3}{y}^{3}\hfill \end{array}$ |

**Theorem**

**8.**

- 1
- ${M}_{1}(G)=680\times {2}^{n}-622$
- 2
- ${M}_{2}(G)=836\times {2}^{n}-771$
- 3
- ${}^{m}{M}_{2}(G)=26\times {2}^{n}-\frac{209}{9}$
- 4
- ${R}_{\alpha}(G)=({4}^{\alpha}\times 56+{6}^{\alpha}\times 48+{9}^{\alpha}\times 36)\times {2}^{n}-({4}^{\alpha}\times 48+{6}^{\alpha}\times 44+{9}^{\alpha}\times 35)$
- 5
- ${R}_{\alpha}(G)=(\frac{1}{{4}^{\alpha}}\times 56+\frac{1}{{6}^{\alpha}}\times 48+\frac{1}{{9}^{\alpha}}\times 36)\times {2}^{n}-(\frac{1}{{4}^{\alpha}}\times 48+\frac{1}{{6}^{\alpha}}\times 44+\frac{1}{{9}^{\alpha}}\times 35)$
- 6
- $SDD(G)=196\times {2}^{n}-\frac{535}{3}$

**Proof.**

**Theorem**

**9.**

- 1
- ${M}_{1}(G)=9mn-5m$
- 2
- ${M}_{2}(G)=\frac{27}{2}mn-\frac{21}{2}m$
- 3
- ${}^{m}{M}_{2}(G)=\frac{1}{3}m+\frac{m}{18}(3n-5)$
- 4
- ${R}_{\alpha}(G)={6}^{\alpha}\times 2m+{9}^{\alpha}(\frac{1}{2}m(3n-5))$
- 5
- ${R}_{\alpha}(G)=\frac{1}{{6}^{\alpha}}\times 2m+\frac{1}{{9}^{\alpha}}(\frac{1}{2}m(3n-5))$
- 6
- $SDD(G)=\frac{13}{3}m+m(3n-5)$

**Proof.**

$\begin{array}{cc}\hfill {D}_{x}(g(x,y))& =4m{x}^{2}{y}^{3}+\frac{3}{2}m(3n-5){x}^{3}{y}^{3}\hfill \\ \hfill {D}_{y}(g(x,y))& =6m{x}^{2}{y}^{3}+\frac{3}{2}m(3n-5){x}^{3}{y}^{3}\hfill \\ \hfill {S}_{x}(g(x,y))& =m{x}^{2}{y}^{3}+\frac{m}{6}(3n-5){x}^{3}{y}^{3}\hfill \\ \hfill {S}_{y}(g(x,y))& =\frac{2m}{3}{x}^{2}{y}^{3}+\frac{m}{6}(3n-5){x}^{3}{y}^{3}\hfill \end{array}$ |

$\begin{array}{cc}\hfill {D}_{x}{D}_{y}(g(x,y))& =12m{x}^{2}{y}^{3}+\frac{9}{2}m(3n-5){x}^{3}{y}^{3}\hfill \\ \hfill {S}_{x}{S}_{y}(g(x,y))& =\frac{m}{3}{x}^{2}{y}^{3}+\frac{m}{18}(3n-5){x}^{3}{y}^{3}\hfill \\ \hfill {D}_{x}{S}_{y}(g(x,y))& =\frac{4}{3}m{x}^{2}{y}^{3}+\frac{m}{2}(3n-5){x}^{3}{y}^{3}\hfill \\ \hfill {D}_{y}{S}_{x}(g(x,y))& =3m{x}^{2}{y}^{2}+\frac{m}{2}(3n-5){x}^{3}{y}^{3}\hfill \end{array}$ |

$\begin{array}{cc}\hfill {D}_{y}^{\alpha}(g(x,y))& ={3}^{\alpha}\times 2m{x}^{2}{y}^{3}+{3}^{\alpha}(\frac{m}{2}(3n-5)){x}^{3}{y}^{3}\hfill \\ \hfill {S}_{y}^{\alpha}(g(x,y))& =\frac{1}{{3}^{\alpha}}\times 2m{x}^{2}{y}^{3}+\frac{1}{{3}^{\alpha}}(\frac{m}{2}(3n-5)){x}^{3}{y}^{3}\hfill \end{array}$ |

$\begin{array}{cc}\hfill {D}_{x}^{\alpha}{D}_{y}^{\alpha}(g(x,y))& ={6}^{\alpha}\times 2m{x}^{2}{y}^{3}+{9}^{\alpha}(\frac{m}{2}(3n-5)){x}^{3}{y}^{3}\hfill \\ \hfill {S}_{x}^{\alpha}{S}_{y}^{\alpha}(g(x,y))& =\frac{1}{{6}^{\alpha}}\times 2m{x}^{2}{y}^{3}+\frac{1}{{9}^{\alpha}}(\frac{m}{2}(3n-5)){x}^{3}{y}^{3}\hfill \end{array}$ |

**Theorem**

**10.**

- 1
- ${M}_{1}(G)=9mn-10m$
- 2
- ${M}_{2}(G)=\frac{27}{2}mn-20m$
- 3
- ${}^{m}{M}_{2}(G)=\frac{1}{6}mn+\frac{5}{36}m$
- 4
- ${R}_{\alpha}(G)={4}^{\alpha}\times m+{6}^{\alpha}\times 2m+{9}^{\alpha}\frac{1}{2}(m)(3n-8)$
- 5
- ${R}_{\alpha}(G)=\frac{1}{{4}^{\alpha}}\times m+\frac{1}{{6}^{\alpha}}\times 2m+\frac{1}{{9}^{\alpha}}\frac{1}{2}(m)(3n-8)$
- 6
- $SDD(G)=\frac{25}{6}m+m(3n-8)$

**Proof.**

**Theorem**

**11.**

- 1
- ${M}_{1}(G)=36mn-40m$
- 2
- ${M}_{2}(G)=108mn-160m$
- 3
- ${}^{m}{M}_{2}(G)=\frac{7}{24}m+\frac{1}{36}(3mn-8m)$
- 4
- ${R}_{\alpha}(G)={16}^{\alpha}\times 2m+{24}^{\alpha}\times 4m+{36}^{\alpha}\times (3mn-8m)$
- 5
- ${R}_{\alpha}(G)=\frac{1}{{16}^{\alpha}}\times 2m+\frac{1}{{24}^{\alpha}}\times 4m+\frac{1}{{36}^{\alpha}}\times (3mn-8m)$
- 6
- $SDD(G)=6mn-\frac{16}{3}m$

**Theorem**

**12.**

- 1
- ${M}_{1}(G)=\frac{74}{3}mn-\frac{52}{3}m$
- 2
- ${M}_{2}(G)=65mn-\frac{208}{3}m$
- 3
- ${}^{m}{M}_{2}(G)=\frac{19}{225}mn+\frac{133}{1800}m$
- 4
- ${R}_{\alpha}(G)=({30}^{\alpha}+{25}^{\alpha})mn+({16}^{\alpha}+\frac{8}{3}\times {20}^{\alpha}+\frac{4}{3}\times {24}^{\alpha}-\frac{8}{3}\times {25}^{\alpha}+4\times {30}^{\alpha})m$
- 5
- ${R}_{\alpha}(G)=(\frac{1}{{30}^{\alpha}}+\frac{1}{{25}^{\alpha}})mn+(\frac{1}{{16}^{\alpha}}+\frac{8}{3}\times \frac{1}{{20}^{\alpha}}+\frac{4}{3}\times \frac{1}{{24}^{\alpha}}-\frac{8}{3}\times \frac{1}{{25}^{\alpha}}+4\times \frac{1}{{30}^{\alpha}})m$
- 6
- $SDD(G)=\frac{212}{45}mn-\frac{10}{9}m$

## 4. Discussions

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**(

**A**) Molecular graph of $N{D}_{1}\left[2\right]$; (

**B**) molecular graph of $N{D}_{2}\left[2\right]$.

**Figure 7.**Comparison of some of the indices of $N{D}_{1}\left[n\right]$, $D2\left[n\right]$ for n = 0,...,10.

Topological Index | $\mathit{g}(\mathit{x},\mathit{y})$ | Derivation from $\mathit{M}(\mathit{G},\mathit{x},\mathit{y})$ |
---|---|---|

${M}_{1}(G)$ | $x+y$ | ${\left[({D}_{x}+{D}_{y})(M(G,x,y))\right]}_{(x,y)=(1,1)}$ |

${M}_{2}(G)$ | $xy$ | ${\left[({D}_{x}{D}_{y})(M(G,x,y))\right]}_{(x,y)=(1,1)}$ |

${}^{m}{M}_{2}(G)$ | $\frac{1}{xy}$ | ${\left[({S}_{x}{S}_{y})(M(G,x,y))\right]}_{(x,y)=(1,1)}$ |

${R}_{\alpha}(G)$ | ${(xy)}^{\alpha}$ | ${\left[({D}_{x}^{\alpha}{D}_{y}^{\alpha})(M(G,x,y))\right]}_{(x,y)=(1,1)}$ |

${R}_{\alpha}(G)$ | $\frac{1}{{(xy)}^{\alpha}}$ | ${\left[({S}_{x}^{\alpha}{S}_{y}^{\alpha})(M(G,x,y))\right]}_{(x,y)=(1,1)}$ |

$SDD(G)$ | $\frac{{x}^{2}+{y}^{2}}{xy}$ | ${\left[({S}_{x}{D}_{y}+{S}_{y}{D}_{x})(M(G,x,y))\right]}_{(x,y)=(1,1)}$ |

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**MDPI and ACS Style**

Raza, Z.; Essa K. Sukaiti, M.
M-Polynomial and Degree Based Topological Indices of Some Nanostructures. *Symmetry* **2020**, *12*, 831.
https://doi.org/10.3390/sym12050831

**AMA Style**

Raza Z, Essa K. Sukaiti M.
M-Polynomial and Degree Based Topological Indices of Some Nanostructures. *Symmetry*. 2020; 12(5):831.
https://doi.org/10.3390/sym12050831

**Chicago/Turabian Style**

Raza, Zahid, and Mark Essa K. Sukaiti.
2020. "M-Polynomial and Degree Based Topological Indices of Some Nanostructures" *Symmetry* 12, no. 5: 831.
https://doi.org/10.3390/sym12050831