Some New Properties for Degree-Based Graph Entropies

Guoxiang Lu,Lijia Wang,Bingqing Li

doi:10.3390/e17127871

Guoxiang Lu, Lijia Wang + Show 1 more

Open Access

https://doi.org/10.3390/e17127871

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Abstract

The graph entropies inspired by Shannon’s entropy concept become the information-theoretic quantities for measuring the structural information of graphs and complex networks. In this paper, we continue studying some new properties of the graph entropies based on information functionals involving vertex degrees. We prove the monotonicity of the graph entropies with respect to the power exponent. Considering only the maximum and minimum degrees of the ( n , m ) -graph, we obtain some upper and lower bounds for the degree-based graph entropy. These bounds have different performances to restrict the degree-based graph entropy in different kinds of graphs. Moreover the degree-based graph entropy can be estimated by these bounds.

Highlights

The entropy of a probability distribution is known as a measure of the unpredictability of information content or a measure of the uncertainty of a system
Because the degrees of the graph G can be seen as one species of the most noticeable invariants and they can be calculated very in large-scale networks, we focus on the information functionals by using the degree powers and the graph entropies based on the degrees
By calculating the graph entropy of a series of path graphs, we show some values of the degree-based graph entropy and bounds in the following Table 1

Summary

Introduction

The entropy of a probability distribution is known as a measure of the unpredictability of information content or a measure of the uncertainty of a system. This concept was introduced first from Shannon’s famous paper [1]. Entropy was initiated to be applied to graphs It was developed for measuring the structural information of graphs and networks [2]. For intrinsic graph entropy measures, the probability distribution is induced by some structural feature of the graph. The structure of this paper is as follows: In Section 2, some definitions and notations of graph theory and the graph entropies we are going to study are reviewed. A short summary and conclusion are drawn in the last section

Preliminaries to Degree-Based Graph Entropies

Path Graph

Star Graph

Monocentric Homogeneous Dendrimer Graph

Summary and Conclusions