# Some Invariants of Jahangir Graphs

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## Abstract

**:**

_{n},

_{m}for all values of m and n. From the M-polynomial, we recover many degree-based topological indices such as first and second Zagreb indices, modified Zagreb index, Symmetric division index, etc. We also compute harmonic index, first and second multiple Zagreb indices and forgotten index of Jahangir graphs. Our results are extensions of many existing results.

## 1. Introduction

_{n},

_{m}is a graph on $nm+1$ vertices and $m(n+1)$ edges $\forall \text{}n\ge 2$ and $m\ge 3$. J

_{n},

_{m}consists of a cycle ${c}_{nm}$ with one additional vertex which is adjacent to m vertices of ${c}_{nm}$ at distance to each other. Figure 1 shows some particular cases of J

_{n},

_{m}.

_{n},

_{m}. We also compute many degree-based topological indices for this family of the graph. We analyze these indices against parametric values m and n graphically and draw some nice conclusions as well.

**Definition**

**1.**

**Definition**

**2.**

**Definition**

**3.**

**Definition**

**4.**

**Definition**

**5.**

**Definition**

**6.**

**Definition**

**7.**

## 2. Main Results

**Theorem**

**1.**

**Proof.**

**Proposition**

**2.**

- 1
- ${M}_{1}({J}_{n,m})={m}^{2}+4mn+5m,$
- 2
- ${M}_{2}({J}_{n,m})=3{m}^{2}+4mn+4m,$
- 3
- ${}^{m}{M}_{2}({J}_{n,m})=(1/4)mn-(1/6)m+1/3,$
- 4
- ${R}_{\alpha}\left({J}_{n,m}\right)={4}^{a}m(n-2)+2m{6}^{a}+{m}^{(a+1)}{3}^{a},$
- 5
- $R{R}_{\alpha}\left({J}_{n,m}\right)=m(n-2){4}^{(-a)}+2m{6}^{(-a)}+{m}^{(1-a)}{3}^{(-a)},$
- 6
- $SSD({J}_{n,m})=2mn+(1/3)m+3+(1/3){m}^{2},$
- 7
- $H\left({J}_{n,m}\right)=(1/4)mn-(1/10)m+m/(m+3),$
- 8
- $I({J}_{n,m})=mn+(2/5)m+3{m}^{2}/(m+3),$
- 9
- $A({J}_{n,m})=8mn-8m+27{m}^{4}/{(1+m)}^{3}.$

**Proof.**

**1. First Zagreb Index**

**2. Second Zagreb Index**

**3. Modified second Zagreb Index**

**4. Generalized Randic Index**

**5. Inverse Randic Index**

**6. Symmetric Division Index**

**7. Harmonic Index**

**8. Inverse Sum Index**

**9. Augmented Zagreb Index**

**Theorem**

**3.**

**Proof.**

**Proposition**

**4.**

**Proof.**

## 3. Conclusions and Discussion

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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

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

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

Second Modified Zagreb | $\left({S}_{x}{S}_{y}\right){(M(G;x,y))}_{x=y=1}$ |

Inverse Randic | $\left({D}_{x}^{\alpha}{D}_{y}^{\alpha}\right){(M(G;x,y))}_{x=y=1}$ |

General Randic | $\left({S}_{x}^{\alpha}{S}_{y}^{\alpha}\right){(M(G;x,y))}_{x=y=1}$ |

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

Harmonic Index | $2{S}_{x}J{(M(G;x,y))}_{x=1}$ |

Inverse sum Index | ${S}_{x}J{D}_{x}{D}_{y}{(M(G;x,y))}_{x=1}$ |

Augmented Zagreb Index | ${{S}_{x}}^{3}{Q}_{-2}J{{D}_{x}}^{3}{{D}_{y}}^{3}{(M(G;x,y))}_{x=1}$ |

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**MDPI and ACS Style**

Munir, M.; Nazeer, W.; Kang, S.M.; Qureshi, M.I.; Nizami, A.R.; Kwun, Y.C. Some Invariants of Jahangir Graphs. *Symmetry* **2017**, *9*, 17.
https://doi.org/10.3390/sym9010017

**AMA Style**

Munir M, Nazeer W, Kang SM, Qureshi MI, Nizami AR, Kwun YC. Some Invariants of Jahangir Graphs. *Symmetry*. 2017; 9(1):17.
https://doi.org/10.3390/sym9010017

**Chicago/Turabian Style**

Munir, Mobeen, Waqas Nazeer, Shin Min Kang, Muhammad Imran Qureshi, Abdul Rauf Nizami, and Youl Chel Kwun. 2017. "Some Invariants of Jahangir Graphs" *Symmetry* 9, no. 1: 17.
https://doi.org/10.3390/sym9010017