# Wiener Complexity versus the Eccentric Complexity

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## Abstract

**:**

## 1. Introduction

**Theorem**

**1.**

**Problem**

**1.**

## 2. Trees

**Lemma**

**1.**

**Proof.**

**Lemma**

**2.**

**Proof.**

**Theorem**

**2.**

- (i)
- If $3\le {c}_{1}\le {c}_{2}$ then there are infinitely many trees T with ${C}_{\mathrm{ec}}\left(T\right)={c}_{1}$ and ${C}_{W}\left(T\right)={c}_{2}$.
- (ii)
- If ${c}_{1}=2$ and ${c}_{2}\in \{2,4\}$ then there are infinitely many trees T with ${C}_{\mathrm{ec}}\left(T\right)={c}_{1}$ and ${C}_{W}\left(T\right)={c}_{2}$ and no trees with ${C}_{\mathrm{ec}}\left(T\right)={c}_{1}$ and ${C}_{W}\left(T\right)\notin \{2,4\}$.
- (iii)
- If ${c}_{1}=1$ then there are only two trees T with ${C}_{\mathrm{ec}}\left(T\right)=1$ and in this case ${C}_{W}\left(T\right)=1$ as well.

**Proof.**

**Corollary**

**1.**

## 3. Unicyclic Graphs

**Lemma**

**3.**

**Proof.**

**Theorem**

**3.**

**Proof.**

**Corollary**

**2.**

## 4. Graphs with Diameter 3

**Lemma**

**4.**

**Proof.**

**Proposition**

**1.**

**Proof.**

**Lemma**

**5.**

**Proof.**

**Theorem**

**4.**

**Proof.**

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 3.**The unicyclic graph on 13 vertices and with odd cycle that has Wiener complexity smaller than eccentric complexity.

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Knor, M.; Škrekovski, R.
Wiener Complexity versus the Eccentric Complexity. *Mathematics* **2021**, *9*, 79.
https://doi.org/10.3390/math9010079

**AMA Style**

Knor M, Škrekovski R.
Wiener Complexity versus the Eccentric Complexity. *Mathematics*. 2021; 9(1):79.
https://doi.org/10.3390/math9010079

**Chicago/Turabian Style**

Knor, Martin, and Riste Škrekovski.
2021. "Wiener Complexity versus the Eccentric Complexity" *Mathematics* 9, no. 1: 79.
https://doi.org/10.3390/math9010079