# On M-Polynomials of Dunbar Graphs in Social Networks

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## Abstract

**:**

## 1. Introduction

## 2. Defining Network Structure as M-Polynomial

#### 2.1. The $(m,n,r)$-Agent Recruitment Graph

**Definition**

**1**

**Definition**

**2.**

#### 2.2. Topological Indices From The M-polynomial

**Theorem**

**1.**

**Proof.**

**Theorem**

**2.**

**Proof.**

**Theorem**

**3.**

**Proof.**

**Theorem**

**4.**

**Proof.**

**Theorem**

**5.**

**Proof.**

**Theorem**

**6.**

**Proof.**

**Theorem**

**7.**

**Proof.**

**Theorem**

**8.**

**Proof.**

**Theorem**

**9.**

**Proof.**

## 3. Dunbar Graphs and Topological Indices

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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Topological Index | Notion for Topological Index | Derivation from $\mathit{M}(\mathit{G},\mathit{x},\mathit{y})$ or $\mathit{f}(\mathit{x},\mathit{y})$ |
---|---|---|

First Zagreb | ${M}_{1}\left(G\right)$ | $({D}_{x}+{D}_{y})\left(f(x,y)\right){\mid}_{x=y=1}$ |

Second Zagreb | ${M}_{2}\left(G\right)$ | (${D}_{x}{D}_{y})\left(f(x,y)\right){\mid}_{x=y=1}$ |

Second modified Zagreb | ${}^{m}$${M}_{2}\left(G\right)$ | ${S}_{x}{S}_{y}\left(f(x,y)\right){\mid}_{x=y=1}$ |

General Randi$\stackrel{\xb4}{c}$ | ${R}_{\alpha}\left(G\right)$ | ${D}_{x}^{\alpha}{D}_{y}^{\alpha}\left(f(x,y)\right){\mid}_{x=y=1}$ |

General Inverse Randi$\stackrel{\xb4}{c}$ | $R{R}_{\alpha}\left(G\right)$ | ${S}_{x}^{\alpha}{S}_{y}^{\alpha}\left(f(x,y)\right){\mid}_{x=y=1}$ |

Symmetric Division Index | $SDD\left(G\right)$ | $({D}_{x}{S}_{y}+{D}_{y}{S}_{x})\left(f(x,y)\right){\mid}_{x=y=1}$ |

Harmonic Index | H(G) | 2${S}_{x}J\left(f(x,y)\right){\mid}_{x=1}$ |

Inverse sum Index | I(G) | ${S}_{x}J{D}_{x}{D}_{y}\left(f(x,y)\right){\mid}_{x=1}$ |

Augmented Zagreb Index | AZI(G) | ${S}_{x}^{3}{Q}_{-2}J{D}_{x}^{3}{D}_{y}^{3}\left(f(x,y)\right){\mid}_{x=1}$ |

${\mathit{d}}_{\mathit{u}}$ | 1 | m | $(\mathit{n}+1)$ |
---|---|---|---|

Number of vertices | $m{n}^{r-1}$ | 1 | $\frac{m({n}^{r-1}-1)}{(n-1)}$ |

$({\mathit{d}}_{\mathit{u}},{\mathit{d}}_{\mathit{v}})$ | (1, $\mathit{n}+1$) | (m, $\mathit{n}+1$) | ($\mathit{n}+1$, $\mathit{n}+1$) |
---|---|---|---|

Number of edges | $m{n}^{r-1}$ | m | $\frac{mn({n}^{r-2}-1)}{(n-1)}$ |

$(\mathit{m},\mathit{n},\mathit{r})$ | ${\mathit{M}}_{1}\left(\mathcal{G}\right)$ | ${\mathit{M}}_{2}\left(\mathcal{G}\right)$ | ${}^{\mathit{m}}$${\mathit{M}}_{2}\left(\mathcal{G}\right)$ | $\mathit{SDD}\left(\mathcal{G}\right)$ | $\mathit{H}\left(\mathcal{G}\right)$ | $\mathit{I}\left(\mathcal{G}\right)$ | $\mathit{AZI}\left(\mathcal{G}\right)$ |
---|---|---|---|---|---|---|---|

(5,5,6) | 156,230 | 234,300 | 2712.6667 | 96,364.333 | 5115.1948 | 5115.1948 | 13,406.494 |

(5,150,6) | 5.848$\times {10}^{13}$ | 1.154$\times {10}^{14}$ | 2.515$\times {10}^{9}$ | 5.734$\times {10}^{13}$ | 5.013$\times {10}^{9}$ | 3.772$\times {10}^{11}$ | 3.873$\times {10}^{11}$. |

(5,5,4) | 6230 | 9300 | 108.5 | 3864.3333 | 204.48052 | 549.35065 | 1265.1852 |

(5,150,4) | 2.599$\times {10}^{9}$ | 5.130$\times {10}^{9}$ | 111759.94 | 2.548$\times {10}^{9}$ | 222,789.54 | 16,764,004 | 17,215,344 |

(150,150,6) | 1.754$\times {10}^{15}$ | 3.463$\times {10}^{15}$ | 7.544$\times {10}^{10}$ | 1.720$\times {10}^{15}$ | 1.504$\times {10}^{11}$ | 1.132$\times {10}^{13}$ | 1.162$\times {10}^{13}$ |

(150,150,4) | 7.798$\times {10}^{10}$ | 1.539$\times {10}^{11}$ | 3352798 | 7.645$\times {10}^{10}$ | 6,683,685.2 | 5.029$\times {10}^{8}$ | 5.816$\times {10}^{8}$ |

(150,5,6) | 470,8650 | 715,9500 | 81,375.167 | 2,894,381 | 153,430.49 | 402,651.1 | 839,940.33 |

(150,5,4) | 208,650 | 409,500 | 3250.1667 | 11,9381 | 6109.0659 | 16,936.813 | 62,340.333 |

(5,15,6) | 73,225,380 | 1.302$\times {10}^{8}$ | 238,364.12 | 60,987,322 | 463,641.65 | 357,3548.5 | 4,608,373.2 |

(5,15,4) | 325,380 | 577,600 | 1059.4375 | 271,072 | 2060.7703 | 15,901.401 | 20,853.232 |

(15,5,6) | 468,840 | 703,800 | 8137.6667 | 289,106 | 15,344.286 | 40,242.857 | 82,594.256 |

(15,5,4) | 18,840 | 28,800 | 325.16667 | 11,606 | 612.14286 | 1671.4286 | 4834.2557 |

(15,15,6) | 2.197$\times {10}^{8}$ | 3.905$\times {10}^{8}$ | 715,092.25 | 1.830$\times {10}^{8}$ | 1,390,924.5 | 10,720,704 | 13,832,502 |

(15,15,4) | 976,290 | 1,735,200 | 3178.1875 | 813,194.12 | 6181.8501 | 47,763.188 | 69,942.194 |

(15,150,6) | 1.754$\times {10}^{14}$ | 3.463$\times {10}^{14}$ | 7.544$\times {10}^{9}$ | 1.720$\times {10}^{14}$ | 1.504$\times {10}^{10}$ | 1.132$\times {10}^{12}$ | 1.162$\times {10}^{12}$. |

(15,150,4) | 7.798$\times {10}^{9}$ | 1.539$\times {10}^{10}$ | 35,279.81 | 7.645$\times {10}^{9}$ | 668,368.6 | 50,292,145 | 51,683,780 |

${\mathit{R}}_{\mathit{\alpha}}\left(\mathcal{G}\right)$ | ${\mathit{RR}}_{\mathit{\alpha}}\left(\mathcal{G}\right)$ | ||||||

15,625$\times {6}^{\alpha}$ + 5${}^{(1+\alpha )}\times $6${}^{\alpha}$ + 3900$\times {6}^{2\alpha}$ | 15625$\times {6}^{-\alpha}$ + 5${}^{(1-\alpha )}\times $6${}^{-\alpha}$ + 3900$\times {6}^{-2\alpha}$ | ||||||

3.797$\times {10}^{11}$$\times {151}^{\alpha}$ + 5${}^{(1+\alpha )}\times $151${}^{\alpha}$ + 2.548$\times {10}^{9}$ $\times {151}^{2\alpha}$ | 3.797$\times {10}^{11}$$\times {151}^{-\alpha}$ + 5${}^{(1-\alpha )}\times $151${}^{-\alpha}$ + 2.548$\times {10}^{9}$ $\times {151}^{-2\alpha}$ | ||||||

625$\times {6}^{\alpha}$ + 5${}^{(1+\alpha )}\times $6${}^{\alpha}$ + 150$\times {6}^{2\alpha}$ | 625$\times {6}^{-\alpha}$ + 5${}^{(1-\alpha )}\times $6${}^{-\alpha}$ + 150$\times {6}^{-2\alpha}$ | ||||||

16,875,000$\times {151}^{\alpha}$ + 5${}^{(1+\alpha )}\times $151${}^{\alpha}$ + 113,250$\times {151}^{2\alpha}$ | 16,875,000$\times {151}^{-\alpha}$ + 5${}^{(1-\alpha )}\times $151${}^{-\alpha}$ + 113,250$\times {151}^{-2\alpha}$ | ||||||

1.139$\times {10}^{13}$$\times {151}^{\alpha}$ + 150${}^{(1+\alpha )}\times $151${}^{\alpha}$ + 7.645$\times {10}^{10}$ $\times {151}^{2\alpha}$ | 1.139$\times {10}^{13}$$\times {151}^{-\alpha}$ + 150${}^{(1-\alpha )}\times $151${}^{-\alpha}$ + 7.645$\times {10}^{10}$ $\times {151}^{-2\alpha}$ | ||||||

5.062$\times {10}^{8}$$\times {151}^{\alpha}$ + 150${}^{(1+\alpha )}\times $151${}^{\alpha}$ + 3,397,500 $\times {151}^{2\alpha}$ | 5.062$\times {10}^{8}$$\times {151}^{-\alpha}$ + 150${}^{(1-\alpha )}\times $151${}^{-\alpha}$ + 3,397,500 $\times {151}^{-2\alpha}$ | ||||||

468,750$\times {6}^{\alpha}$ + 150${}^{(1+\alpha )}\times $6${}^{\alpha}$ + 117,000 $\times {6}^{2\alpha}$ | 468,750$\times {6}^{-\alpha}$ + 150${}^{(1-\alpha )}\times $6${}^{-\alpha}$ + 117,000 $\times {6}^{-2\alpha}$ | ||||||

18,750$\times {6}^{\alpha}$ + 150${}^{(1+\alpha )}\times $6${}^{\alpha}$ + 4500 $\times {6}^{2\alpha}$ | 18,750$\times {6}^{-\alpha}$ + 150${}^{(1-\alpha )}\times $6${}^{-\alpha}$ + 4500 $\times {6}^{-2\alpha}$ | ||||||

3796875$\times {16}^{\alpha}$ + 5${}^{(1+\alpha )}\times $16${}^{\alpha}$ + 271,200 $\times {16}^{2\alpha}$ | 3,796,875$\times {16}^{-\alpha}$ + 5${}^{(1-\alpha )}\times $16${}^{-\alpha}$ + 271,200 $\times {16}^{-2\alpha}$ | ||||||

16,875$\times {16}^{\alpha}$ + 5${}^{(1+\alpha )}\times $16${}^{\alpha}$ + 1200 $\times {16}^{2\alpha}$ | 16,875$\times {16}^{-\alpha}$ + 5${}^{(1-\alpha )}\times $16${}^{-\alpha}$ + 1200 $\times {16}^{-2\alpha}$ | ||||||

46,875$\times {6}^{\alpha}$ + 15${}^{(1+\alpha )}\times $6${}^{\alpha}$ + 11,700 $\times {6}^{2\alpha}$ | 46,875$\times {6}^{-\alpha}$ + 15${}^{(1-\alpha )}\times $6${}^{-\alpha}$ + 11,700 $\times {6}^{-2\alpha}$ | ||||||

1875$\times {6}^{\alpha}$ + 15${}^{(1+\alpha )}\times $6${}^{\alpha}$ + 450 $\times {6}^{2\alpha}$ | 1875$\times {6}^{-\alpha}$ + 15${}^{(1-\alpha )}\times $6${}^{-\alpha}$ + 450 $\times {6}^{-2\alpha}$ | ||||||

11,390,625$\times {6}^{\alpha}$ + 15${}^{(1+\alpha )}\times $16${}^{\alpha}$ + 813,600 $\times {16}^{2\alpha}$ | 11,390,625$\times {6}^{-\alpha}$ + 15${}^{(1-\alpha )}\times $16${}^{-\alpha}$ + 813,600 $\times {16}^{-2\alpha}$ | ||||||

50,625$\times {6}^{\alpha}$ + 15${}^{(1+\alpha )}\times $16${}^{\alpha}$ + 3600 $\times {16}^{2\alpha}$ | 50,625$\times {6}^{-\alpha}$ + 15${}^{(1-\alpha )}\times $16${}^{-\alpha}$ + 3600 $\times {16}^{-2\alpha}$ | ||||||

1.139$\times {10}^{12}$$\times {151}^{\alpha}$ + 15${}^{(1+\alpha )}\times $151${}^{\alpha}$ + 7.645$\times {10}^{9}$$\times {151}^{2\alpha}$ | 1.139$\times {10}^{12}$$\times {151}^{-\alpha}$ + 15${}^{(1-\alpha )}\times $151${}^{-\alpha}$ + 7.645$\times {10}^{9}$$\times {151}^{-2\alpha}$, | ||||||

50,625,000$\times {151}^{\alpha}$ + 15${}^{(1+\alpha )}\times $151${}^{\alpha}$ + 339,750$\times {151}^{2\alpha}$ | 50,625,000$\times {151}^{-\alpha}$ + 15${}^{(1-\alpha )}\times $151${}^{-\alpha}$ + 339,750$\times {151}^{-2\alpha}$, |

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**MDPI and ACS Style**

Acharjee, S.; Bora, B.; Dunbar, R.I.M.
On M-Polynomials of Dunbar Graphs in Social Networks. *Symmetry* **2020**, *12*, 932.
https://doi.org/10.3390/sym12060932

**AMA Style**

Acharjee S, Bora B, Dunbar RIM.
On M-Polynomials of Dunbar Graphs in Social Networks. *Symmetry*. 2020; 12(6):932.
https://doi.org/10.3390/sym12060932

**Chicago/Turabian Style**

Acharjee, Santanu, Bijit Bora, and Robin I. M. Dunbar.
2020. "On M-Polynomials of Dunbar Graphs in Social Networks" *Symmetry* 12, no. 6: 932.
https://doi.org/10.3390/sym12060932