# Some Bicyclic Graphs Having 2 as Their Laplacian Eigenvalues

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## Abstract

**:**

## 1. Introduction

## 2. Preliminary Results

**Theorem**

**1**

**.**The Laplacian coefficient ${\xi}_{n-k}$ of a graph G of order n is given by ${\xi}_{n-k}={\sum}_{F\in {\mathfrak{F}}_{k}}\gamma \left(F\right)$, where ${\mathfrak{F}}_{k}$ is the set of all spanning forest of G with exactly k components.

## 3. The Eigenvector of the Laplacian Eigenvalue 2

**Theorem**

**2**

**.**Let G be a graph on n vertices and e be an edge of G. Let ${\lambda}_{i}\left(G\right),\phantom{\rule{4pt}{0ex}}(1\le i\le n)$ be the eigenvalues of $L\left(G\right)$. Then the following holds:

**Remark**

**1.**

**Theorem**

**3.**

**Proof.**

**Theorem**

**4.**

**Proof.**

## 4. The Laplacian Eigenvalue 2 of Bicyclic Graphs

**Lemma**

**2.**

**Proof.**

**Theorem**

**5.**

**Proof.**

**Remark**

**3.**

**Theorem**

**6.**

**Proof.**

**Corollary**

**4.**

**Theorem**

**7.**

**Proof.**

**Corollary**

**5.**

**Example**

**6.**

**Theorem**

**8.**

**Proof.**

**Theorem**

**9.**

**Proof.**

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**g = 3; ${x}_{T}\left(u\right)={x}_{T}\left(v\right)$; bold edges represent those in the perfect matching M.

**Figure 3.**g = 5; ${x}_{T}\left(u\right)={x}_{T}\left(v\right)$ and ${x}_{{T}^{\prime}}\left({u}^{\prime}\right)={x}_{{T}^{\prime}}\left({v}^{\prime}\right)$.

**Figure 4.**g = 6; ${x}_{T}\left(u\right)={x}_{T}\left(v\right)$, ${x}_{{T}^{\prime}}\left(v\right)={x}_{{T}^{\prime}}\left(w\right)$ and ${x}_{{T}^{\u2033}}\left({u}^{\prime}\right)={x}_{{T}^{\u2033}}\left({v}^{\prime}\right)$.

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**MDPI and ACS Style**

Farkhondeh, M.; Habibi, M.; Mojdeh, D.A.; Rao, Y. Some Bicyclic Graphs Having 2 as Their Laplacian Eigenvalues. *Mathematics* **2019**, *7*, 1233.
https://doi.org/10.3390/math7121233

**AMA Style**

Farkhondeh M, Habibi M, Mojdeh DA, Rao Y. Some Bicyclic Graphs Having 2 as Their Laplacian Eigenvalues. *Mathematics*. 2019; 7(12):1233.
https://doi.org/10.3390/math7121233

**Chicago/Turabian Style**

Farkhondeh, Masoumeh, Mohammad Habibi, Doost Ali Mojdeh, and Yongsheng Rao. 2019. "Some Bicyclic Graphs Having 2 as Their Laplacian Eigenvalues" *Mathematics* 7, no. 12: 1233.
https://doi.org/10.3390/math7121233