# The Characteristic Polynomials of Symmetric Graphs

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## Abstract

**:**

## 1. Introduction

**Theorem**

**1.**

**Example**

**1.**

**Theorem**

**2.**

**Example**

**2.**

## 2. Block Circulant Matrices

**Lemma**

**1.**

**Proof.**

- 1.
- $w({u}_{i},{h}^{m}({v}_{k}^{j}))=w({u}_{i},{v}_{k}^{j})$,
- 2.
- $w({h}^{m}({v}_{{k}^{\prime}}^{0}),{h}^{m}({v}_{k}^{j}))=w({v}_{{k}^{\prime}}^{0},{v}_{k}^{j})$.

## 3. Proofs

${\Delta}_{r+1}=\left[\begin{array}{ccccc}A(F)& {S}^{\prime}& S& \cdots & S\\ {R}^{t}& {A}_{0}^{\prime}& {A}_{1}& \cdots & {A}_{p-1}\\ {R}^{t}& {A}_{p-1}^{\prime}& {A}_{0}& \cdots & {A}_{p-2}\\ \vdots & \vdots & \vdots & \vdots & \vdots \\ {R}^{t}& {A}_{1}^{\prime}& {A}_{2}& \cdots & {A}_{0}\end{array}\right]$, ${\Delta}_{r+s+1}=\left[\begin{array}{ccccc}A(F)& S& {S}^{\prime}& \cdots & S\\ {R}^{t}& {A}_{0}& {A}_{1}^{\prime}& \cdots & {A}_{p-1}\\ {R}^{t}& {A}_{p-1}& {A}_{0}^{\prime}& \cdots & {A}_{p-2}\\ \vdots & \vdots & \vdots & \vdots & \vdots \\ {R}^{t}& {A}_{1}& {A}_{2}^{\prime}& \cdots & {A}_{0}\end{array}\right]$, |

…, ${\Delta}_{r+(p-1)s+1}=\left[\begin{array}{ccccc}A(F)& S& S& \cdots & {S}^{\prime}\\ {R}^{t}& {A}_{0}& {A}_{1}& \cdots & {A}_{p-1}^{\prime}\\ {R}^{t}& {A}_{p-1}& {A}_{0}& \cdots & {A}_{p-2}^{\prime}\\ \vdots & \vdots & \vdots & \vdots & \vdots \\ {R}^{t}& {A}_{1}& {A}_{2}& \cdots & {A}_{0}^{\prime}\end{array}\right]$ |

$\left[\begin{array}{ccccc}A(F)& S& {S}^{\prime}& \cdots & S\\ {R}^{t}& {A}_{1}& {A}_{0}^{\prime}& \cdots & {A}_{p-1}\\ {R}^{t}& {A}_{0}& {A}_{p-1}^{\prime}& \cdots & {A}_{p-2}\\ \vdots & \vdots & \vdots & \vdots & \vdots \\ {R}^{t}& {A}_{2}& {A}_{1}^{\prime}& \cdots & {A}_{0}\end{array}\right].$ |

$\left[\begin{array}{ccccc}A(F)& S& {S}^{\prime}& \cdots & S\\ {R}^{t}& {A}_{2}& {A}_{1}^{\prime}& \cdots & {A}_{0}\\ {R}^{t}& {A}_{1}& {A}_{0}^{\prime}& \cdots & {A}_{p-1}\\ {R}^{t}& {A}_{0}& {A}_{p-1}^{\prime}& \cdots & {A}_{p-2}\\ \vdots & \vdots & \vdots & \vdots & \vdots \\ \Gamma \end{array}\right].$ |

$\left[\begin{array}{ccccc}A(F)& S& {S}^{\prime}& \cdots & S\\ {R}^{t}& {A}_{0}& {A}_{1}^{\prime}& \cdots & {A}_{2}\\ {R}^{t}& {A}_{p-1}& {A}_{0}^{\prime}& \cdots & {A}_{1}\\ {R}^{t}& {A}_{p-2}& {A}_{p-1}^{\prime}& \cdots & {A}_{0}\\ \vdots & \vdots & \vdots & \vdots & \vdots \end{array}\right]$ |

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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## Share and Cite

**MDPI and ACS Style**

Chbili, N.; Al Dhaheri, S.; Tahnon, M.Y.; Abunamous, A.A.E.
The Characteristic Polynomials of Symmetric Graphs. *Symmetry* **2018**, *10*, 582.
https://doi.org/10.3390/sym10110582

**AMA Style**

Chbili N, Al Dhaheri S, Tahnon MY, Abunamous AAE.
The Characteristic Polynomials of Symmetric Graphs. *Symmetry*. 2018; 10(11):582.
https://doi.org/10.3390/sym10110582

**Chicago/Turabian Style**

Chbili, Nafaa, Shamma Al Dhaheri, Mei Y. Tahnon, and Amna A. E. Abunamous.
2018. "The Characteristic Polynomials of Symmetric Graphs" *Symmetry* 10, no. 11: 582.
https://doi.org/10.3390/sym10110582