# Generalized Permanental Polynomials of Graphs

## Abstract

**:**

## 1. Introduction

## 2. Coefficients

#### 2.1. Relation between the Generalized and the Ordinary Permanental Polynomials

**Lemma**

**1.**

**Lemma**

**2.**

**Theorem**

**1.**

**Theorem**

**2.**

#### 2.2. The First Five Coefficients of ${P}_{G}(x,\mu )$

**Lemma**

**3**

**.**Let G be a graph with n vertices and m edges, and let $({d}_{1},{d}_{2},\cdots ,{d}_{n})$ be the degree sequence of G. Suppose that $\pi (A\left(G\right),x)={\sum}_{i=0}^{n}{a}_{i}\left(G\right){x}^{n-i}$. Then,

**Theorem**

**3.**

**Proof.**

**Corollary**

**1.**

**Corollary**

**2.**

#### 2.3. Examples

**Example**

**1.**

**Example**

**2.**

**Example**

**3.**

**Example**

**4.**

**Example**

**5.**

**Example**

**6.**

## 3. Numerical Results

## 4. Conclusions

## Funding

## Conflicts of Interest

## Appendix A

m | # Graphs | # Generalized Perm. Pols | # with Mate | Max. Family |
---|---|---|---|---|

0 | 1 | 1 | 0 | 1 |

1 | 1 | 1 | 0 | 1 |

2 | 2 | 2 | 0 | 1 |

3 | 5 | 5 | 0 | 1 |

4 | 11 | 11 | 0 | 1 |

5 | 26 | 26 | 0 | 1 |

6 | 66 | 66 | 0 | 1 |

7 | 165 | 165 | 0 | 1 |

8 | 428 | 428 | 0 | 1 |

9 | 1103 | 1103 | 0 | 1 |

10 | 2769 | 2768 | 2 | 2 |

11 | 6759 | 6758 | 2 | 2 |

12 | 15,772 | 15,771 | 2 | 2 |

13 | 34,663 | 34,663 | 0 | 1 |

14 | 71,318 | 71,316 | 4 | 2 |

15 | 136,433 | 136,429 | 8 | 2 |

16 | 241,577 | 241,575 | 4 | 2 |

17 | 395,166 | 395,162 | 8 | 2 |

18 | 596,191 | 596,183 | 16 | 2 |

19 | 828,728 | 828,723 | 10 | 2 |

20 | 1,061,159 | 1,061,154 | 10 | 2 |

21 | 1,251,389 | 1,251,381 | 16 | 2 |

22 | 1,358,852 | 1,358,848 | 8 | 2 |

23 | 1,358,852 | 1,358,850 | 4 | 2 |

24 | 1,251,389 | 1,251,385 | 8 | 2 |

25 | 1,061,159 | 1,061,157 | 4 | 2 |

26 | 828,728 | 828,728 | 0 | 1 |

27 | 596,191 | 596,191 | 0 | 1 |

28 | 395,166 | 395,166 | 0 | 1 |

29 | 241,577 | 241,577 | 0 | 1 |

30 | 136,433 | 136,433 | 0 | 1 |

31 | 71,318 | 71,318 | 0 | 1 |

32 | 34,663 | 34,663 | 0 | 1 |

33 | 15,772 | 15,772 | 0 | 1 |

34 | 6759 | 6759 | 0 | 1 |

35 | 2769 | 2769 | 0 | 1 |

36 | 1103 | 1103 | 0 | 1 |

37 | 428 | 428 | 0 | 1 |

38 | 165 | 165 | 0 | 1 |

39 | 66 | 66 | 0 | 1 |

40 | 26 | 26 | 0 | 1 |

41 | 11 | 11 | 0 | 1 |

42 | 5 | 5 | 0 | 1 |

43 | 2 | 2 | 0 | 1 |

44 | 1 | 1 | 0 | 1 |

45 | 1 | 1 | 0 | 1 |

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n | # Graphs | # Generalized Perm. Pols | # with Mate | Frac. with Mate | Max. Family |
---|---|---|---|---|---|

1 | 1 | 1 | 0 | 0 | 1 |

2 | 2 | 2 | 0 | 0 | 1 |

3 | 4 | 4 | 0 | 0 | 1 |

4 | 11 | 11 | 0 | 0 | 1 |

5 | 34 | 34 | 0 | 0 | 1 |

6 | 156 | 156 | 0 | 0 | 1 |

7 | 1044 | 1044 | 0 | 0 | 1 |

8 | 12,346 | 12,346 | 0 | 0 | 1 |

9 | 274,668 | 274,668 | 0 | 0 | 1 |

10 | 12,005,168 | 12,005,115 | 106 | $8.83\times {10}^{-6}$ | 2 |

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**MDPI and ACS Style**

Liu, S.
Generalized Permanental Polynomials of Graphs. *Symmetry* **2019**, *11*, 242.
https://doi.org/10.3390/sym11020242

**AMA Style**

Liu S.
Generalized Permanental Polynomials of Graphs. *Symmetry*. 2019; 11(2):242.
https://doi.org/10.3390/sym11020242

**Chicago/Turabian Style**

Liu, Shunyi.
2019. "Generalized Permanental Polynomials of Graphs" *Symmetry* 11, no. 2: 242.
https://doi.org/10.3390/sym11020242