# Fibonacci Graphs

^{1}

^{2}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Fundamentals and $\Omega $ Invariant

**Observation**

**1.**

**Theorem**

**1.**

**Corollary**

**1.**

**(i)**${F}_{r}+{F}_{r+1}+\cdots +{F}_{k}={F}_{k+2}-{F}_{r+1}$.

**(ii)**${F}_{r+3}-{F}_{r}=2{F}_{r+1}$.

**Definition**

**1.**

**Theorem**

**2.**

**Theorem**

**4.**

**Corollary**

**2.**

## 3. Existence Conditions for Fibonacci Graphs

#### 3.1. Fibonacci Graphs of Order 1

**Theorem**

**5.**

**Proof.**

**Corollary**

**3.**

#### 3.2. Fibonacci Graphs of Order 2

**Theorem**

**6.**

**Corollary**

**4.**

#### 3.3. Fibonacci Graphs of Order 3

**Theorem**

**7.**

**Case**

**1.**

**Case**

**2.**

**Case**

**3.**

**Corollary**

**5.**

#### 3.4. Fibonacci Graphs of Order 4

**Theorem**

**8.**

**Proof.**

#### 3.5. Fibonacci Graphs of Order $N\ge 5$

**Corollary**

**6.**

**(i)**A set$\{{F}_{r},\phantom{\rule{4pt}{0ex}}{F}_{r+1},\phantom{\rule{4pt}{0ex}}{F}_{r+2},\phantom{\rule{4pt}{0ex}}\cdots ,{F}_{r+n-1}\}$consisting of$n=3s+1$consecutive Fibonacci numbers is realizable iff$r\equiv 0\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}modulo\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}3$.

**(ii)**A set$\{{F}_{r},\phantom{\rule{4pt}{0ex}}{F}_{r+1},\phantom{\rule{4pt}{0ex}}{F}_{r+2},\phantom{\rule{4pt}{0ex}}\cdots ,{F}_{r+n-1}\}$consisting of$n=3s+2$consecutive Fibonacci numbers is realizable iff$r\equiv 1\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}modulo\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}3$.

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 10.**The unique connected realization of $D=\left\{{F}_{1}^{\left(1\right)},{F}_{2}^{\left(1\right)},{F}_{3}^{\left(1\right)}\right\}$.

**Figure 11.**The unique disconnected realization of $D=\left\{{F}_{1}^{\left(1\right)},{F}_{2}^{\left(1\right)},{F}_{3}^{\left(1\right)}\right\}$.

**Figure 12.**All three connected realizations of $D=\{{F}_{3k+1}^{\left(1\right)},\phantom{\rule{4pt}{0ex}}{F}_{3k+2}^{\left(1\right)},\phantom{\rule{4pt}{0ex}}{F}_{3k+3}^{\left(1\right)}\}$.

**Figure 13.**The unique disconnected realization of $D=\{{F}_{3k+1}^{\left(1\right)},\phantom{\rule{4pt}{0ex}}{F}_{3k+2}^{\left(1\right)},\phantom{\rule{4pt}{0ex}}{F}_{3k+3}^{\left(1\right)}\}$.

**Figure 14.**All three realizations of $D=\{{F}_{2}^{\left(1\right)},\phantom{\rule{4pt}{0ex}}{F}_{3}^{\left(1\right)},\phantom{\rule{4pt}{0ex}}{F}_{4}^{\left(1\right)}\}$.

**Figure 15.**All three connected realizations of $\{{F}_{3k+2},\phantom{\rule{4pt}{0ex}}{F}_{3k+3},\phantom{\rule{4pt}{0ex}}{F}_{3k+4}\}$.

**Figure 16.**The unique disconnected realization of $\{{F}_{3k+2},\phantom{\rule{4pt}{0ex}}{F}_{3k+3},\phantom{\rule{4pt}{0ex}}{F}_{3k+4}\}$.

**Figure 17.**All three connected realizations of $D=\left\{{F}_{3k+3}^{\left(1\right)},\phantom{\rule{4pt}{0ex}}{F}_{3k+4}^{\left(1\right)},\phantom{\rule{4pt}{0ex}}{F}_{3k+5}^{\left(1\right)}\right\}$.

**Figure 18.**The unique disconnected realization of $D=\left\{{F}_{3k+3}^{\left(1\right)},\phantom{\rule{4pt}{0ex}}{F}_{3k+4}^{\left(1\right)},\phantom{\rule{4pt}{0ex}}{F}_{3k+5}^{\left(1\right)}\right\}$.

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## Share and Cite

**MDPI and ACS Style**

Yurttas Gunes, A.; Delen, S.; Demirci, M.; Cevik, A.S.; Cangul, I.N.
Fibonacci Graphs. *Symmetry* **2020**, *12*, 1383.
https://doi.org/10.3390/sym12091383

**AMA Style**

Yurttas Gunes A, Delen S, Demirci M, Cevik AS, Cangul IN.
Fibonacci Graphs. *Symmetry*. 2020; 12(9):1383.
https://doi.org/10.3390/sym12091383

**Chicago/Turabian Style**

Yurttas Gunes, Aysun, Sadik Delen, Musa Demirci, Ahmet Sinan Cevik, and Ismail Naci Cangul.
2020. "Fibonacci Graphs" *Symmetry* 12, no. 9: 1383.
https://doi.org/10.3390/sym12091383