# Generalized Degree-Based Graph Entropies

## Abstract

**:**

## 1. Introduction

## 2. Preliminaries to Degree-Based Graph Entropy

**Definition**

**1.**

**Definition**

**2.**

**Definition**

**3.**

**Definition**

**4.**

## 3. Generalized Degree-Based Graph Entropy

**Definition**

**5.**

**Definition**

**6.**

**Definition**

**7.**

**Definition**

**8.**

## 4. Properties of the Generalized Degree-Based Graph Entropies

**Proposition**

**1.**

**Proof.**

**Remark**

**1.**

**Proposition**

**2.**

**Proof.**

**Proposition**

**3.**

**Proof.**

**Proposition**

**4.**

**Proof.**

**Remark**

**2.**

**Proposition**

**5.**

**Proof.**

**Theorem**

**1.**

**Proof.**

**Corollary**

**1.**

**Proof.**

**Corollary**

**2.**

**Proof.**

**Theorem**

**2.**

**Proof.**

**Corollary**

**3.**

**Proof.**

**Theorem**

**3.**

**Proof.**

**Corollary**

**4.**

**Proof.**

**Theorem**

**4.**

**Proof.**

## 5. Numerical Results

- (1)
- When $\alpha <1$, the nodes with small value of degree play an important part in the construction of ${I}_{\alpha ,d}^{1}\left(G\right)$ and ${I}_{\alpha ,d}^{2}\left(G\right)$, or they are chosen as the main construction of the complex networks. Especially when the value of $\alpha =0$, each node has the same influence on the whole network from the entropic point of view.
- (2)
- When $\alpha \to 1$, the influence of each node on the network is based on the value of degree for each node. The generalized degree-based graph entropies ${I}_{\alpha ,d}^{1}\left(G\right)$ and ${I}_{\alpha ,d}^{2}\left(G\right)$ are degenerated to the degree-based graph entropy ${I}_{d}\left(G\right)$. So the structure property determined by the node’s degree decides the complexity of the complex network.
- (3)
- When $\alpha >1$, the nodes with big value of degree play an important part in the construction of ${I}_{\alpha ,d}^{1}\left(G\right)$ and ${I}_{\alpha ,d}^{2}\left(G\right)$, or they are chosen as the main construction of the complex networks. The values of the entropies tend to stabilization. The complex network is tended to orderly.

## 6. Summary and Conclusions

## Acknowledgments

## Conflicts of Interest

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**Figure 2.**${I}_{\alpha ,d}^{1}\left(G\right)$ (red), ${I}_{\alpha ,d}^{2}\left(G\right)$ (blue) and ${I}_{d}^{3}\left(G\right)$ (green) versus α. (${I}_{\alpha ,d}^{1}\left(G\right)$, ${I}_{\alpha ,d}^{2}\left(G\right)$ with a pole at $\alpha =1$).

node number | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 |

degree | 3 | 3 | 3 | 2 | 5 | 3 | 5 | 3 | 1 | 4 | 2 |

node number | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | |

degree | 3 | 2 | 2 | 6 | 2 | 3 | 4 | 4 | 3 | 3 |

**Table 2.**The generalized degree-based graph entropies ${I}_{\alpha ,d}^{1}\left(G\right)$ and ${I}_{\alpha ,d}^{2}\left(G\right)$ of the example network.

The Value of α | $-1.0$ | $-0.5$ | 0.0 | 0.5 | $1.0$ | 1.5 | 2.0 | 2.5 | 3.0 | 3.5 | 4.0 |
---|---|---|---|---|---|---|---|---|---|---|---|

${I}_{\alpha ,d}^{1}\left(G\right)$ | 4.505 | 4.447 | 4.392 | 4.342 | $\mathbf{4}.\mathbf{294}$ | 4.249 | 4.206 | 4.165 | 4.127 | 4.090 | 4.056 |

${I}_{\alpha ,d}^{2}\left(G\right)$ | 171.633 | 55.146 | 20.000 | 8.457 | $\mathbf{4}.\mathbf{294}$ | 2.631 | 1.892 | 1.527 | 1.329 | 1.214 | 1.143 |

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Generalized Degree-Based Graph Entropies. *Symmetry* **2017**, *9*, 29.
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Lu G.
Generalized Degree-Based Graph Entropies. *Symmetry*. 2017; 9(3):29.
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