# New Upper Bound and Lower Bound for Degree-Based Network Entropy

^{1}

^{2}

^{3}

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## Abstract

**:**

## 1. Introduction

## 2. Preliminaries to Degree-Based Graph Entropy

**Definition 1.**

**Definition 2.**

**Definition 3.**

## 3. New Upper Bound and Lower Bound for the Degree-Based Network Entropy

**Theorem 1.**

**Proof.**

**Theorem 2.**

**Proof.**

**Theorem 3.**

**Proof.**

**Lemma 1.**

**Proof.**

**Theorem 4.**

**Proof.**

**Remark 1.**

**Theorem 5.**

**Proof.**

**Theorem 6.**

**Proof.**

**Remark 2.**

## 4. Graph Examples and a Practical Network Application

#### 4.1. Monocentric Homogeneous Dendrimer Graph

#### 4.2. Star Graph

#### 4.3. A Special Computer Network

## 5. Summary and Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 4.**The values of $NE$, ${U}_{NE}$ and ${L}_{NE}$ for different nodes. (

**a**) The values of $NE$, ${U}_{NE}$ and ${L}_{NE}$ with different m for 64 nodes; (

**b**) The values of $NE$, ${U}_{NE}$ and ${L}_{NE}$ with different m for 128 nodes; (

**c**) The values of $NE$, ${U}_{NE}$ and ${L}_{NE}$ with different m for 256 nodes; (

**d**) The values of $NE$, ${U}_{NE}$ and ${L}_{NE}$ with different m for 512 nodes.

r | $\mathbf{NE}\left(\mathbf{D}\right(\mathbf{3},\mathbf{r}\left)\right)$ | ${\mathit{U}}_{\mathit{NE}}$ | ${\mathit{L}}_{\mathit{NE}}$ |
---|---|---|---|

1 | 2.0000 | 2.1738 | 1.8734 |

2 | 3.7500 | 4.0780 | 3.7496 |

3 | 5.3928 | 5.7260 | 5.3911 |

4 | 6.9969 | 7.3304 | 6.9936 |

5 | 8.5883 | 8.9217 | 8.5843 |

6 | 10.1754 | 10.5087 | 10.1711 |

7 | 11.7610 | 12.0944 | 11.7567 |

8 | 13.3462 | 13.6796 | 13.3419 |

9 | 14.9313 | 15.2646 | 14.9269 |

10 | 16.5163 | 16.8496 | 16.5119 |

20 | 32.3659 | 32.6993 | 32.3615 |

30 | 48.2155 | 48.5489 | 48.2112 |

40 | 64.0652 | 64.3985 | 64.0608 |

50 | 79.9148 | 80.2481 | 79.9104 |

60 | 95.7644 | 96.0978 | 95.7600 |

70 | 111.6140 | 111.9474 | 111.6097 |

80 | 127.4637 | 127.7970 | 127.4593 |

90 | 143.3133 | 143.6466 | 143.3089 |

100 | 159.1629 | 159.4963 | 159.1585 |

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Lu, G.; Li, B.; Wang, L.
New Upper Bound and Lower Bound for Degree-Based Network Entropy. *Symmetry* **2016**, *8*, 8.
https://doi.org/10.3390/sym8020008

**AMA Style**

Lu G, Li B, Wang L.
New Upper Bound and Lower Bound for Degree-Based Network Entropy. *Symmetry*. 2016; 8(2):8.
https://doi.org/10.3390/sym8020008

**Chicago/Turabian Style**

Lu, Guoxiang, Bingqing Li, and Lijia Wang.
2016. "New Upper Bound and Lower Bound for Degree-Based Network Entropy" *Symmetry* 8, no. 2: 8.
https://doi.org/10.3390/sym8020008