# Orbit Polynomial of Graphs versus Polynomial with Integer Coefficients

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

_{i}indicates the number of orbits of size i in graph G. A new counting polynomial, namely an orbit polynomial, is defined as O

_{G}(x) = ∑

_{i}a

_{i}x

^{i}. Its modified version is obtained by subtracting the orbit polynomial from 1. In the present paper, we studied the conditions under which an integer polynomial can arise as an orbit polynomial of a graph. Additionally, we surveyed graphs with a small number of orbits and characterized several classes of graphs with respect to their orbit polynomials.

## 1. Introduction

## 2. Preliminaries

## 3. Methods and Results

#### 3.1. Orbit Polynomial

**Theorem**

**1.**

- $T\cong {S}_{n}$ if and only if ${O}_{T}\left(x\right)=x+{x}^{n-1}$.
- $T\cong {B}_{\frac{n}{2},\frac{n}{2}}$ if and only if ${O}_{T}\left(x\right)={x}^{2}+{x}^{n-2}$.
- $T\cong {S}_{n,m}$ if and only if ${O}_{T}\left(x\right)=x+{x}^{n}+{x}^{nm}$, where $n=2m$.

**Definition**

**1.**

**Theorem**

**2.**

#### 3.2. Graph Classification with Respect to Orbit Polynomial

**Example**

**1.**

**Theorem**

**3.**

**Proof.**

**Example**

**2.**

**Theorem**

**4.**

**Theorem**

**5.**

**Corollary**

**1.**

**Proof.**

## 4. Integer Polynomials

**Example**

**3.**

**Theorem**

**6.**

**Proof.**

**Theorem**

**7.**

**Proof.**

## 5. Summary and Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 3.**All graphs on six vertices that have three pendant edges, with the orbit polynomial $x+{x}^{2}+{x}^{3}$.

**Figure 6.**Vertices with the same colors are in the same orbit and singleton orbits are shown by white colors.

Partition | ${\mathit{O}}_{\mathit{G}}\left(\mathit{x}\right)$ |
---|---|

6 = 6 | ${x}^{6}$ |

6 = 1 + 5 | $x+{x}^{5}$ |

6 = 1 + 1 + 4 | $2x+{x}^{4}$ |

6 = 2 + 4 | ${x}^{2}+{x}^{4}$ |

6 = 1 + 1 + 1 + 3 | $3x+{x}^{3}$ |

6 = 1 + 2 + 3 | $x+{x}^{2}+{x}^{3}$ |

3 + 3 | $2{x}^{3}$ |

6 = 1 + 1 + 1 + 1 + 2 | $4x+{x}^{2}$ |

6 = 1 + 1 + 2 + 2 | $2x+2{x}^{2}$ |

6 = 2 + 2 + 2 | $3{x}^{2}$ |

6 = 1 + 1 + 1 + 1 + 1 + 1 | $6x$ |

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**MDPI and ACS Style**

Ghorbani, M.; Jalali-Rad, M.; Dehmer, M.
Orbit Polynomial of Graphs versus Polynomial with Integer Coefficients. *Symmetry* **2021**, *13*, 710.
https://doi.org/10.3390/sym13040710

**AMA Style**

Ghorbani M, Jalali-Rad M, Dehmer M.
Orbit Polynomial of Graphs versus Polynomial with Integer Coefficients. *Symmetry*. 2021; 13(4):710.
https://doi.org/10.3390/sym13040710

**Chicago/Turabian Style**

Ghorbani, Modjtaba, Maryam Jalali-Rad, and Matthias Dehmer.
2021. "Orbit Polynomial of Graphs versus Polynomial with Integer Coefficients" *Symmetry* 13, no. 4: 710.
https://doi.org/10.3390/sym13040710