A survey of recent results in (generalized) graph entropies

TL;DR

This survey reviews recent developments in graph entropy measures, exploring their definitions, properties, relationships with graph structures, and applications, highlighting open problems and future research directions.

Contribution

It compiles and analyzes recent results on generalized graph entropies, emphasizing inequalities, relationships, and computational complexity, and discusses applications and open problems.

Findings

Various inequalities and extremal properties established

Relationships between graph entropies and graph energies analyzed

Applications in complexity and open problems discussed

Abstract

The entropy of a graph was first introduced by Rashevsky \cite{Rashevsky} and Trucco \cite{Trucco} to interpret as the structural information content of the graph and serve as a complexity measure. In this paper, we first state a number of definitions of graph entropy measures and generalized graph entropies. Then we survey the known results about them from the following three respects: inequalities and extremal properties on graph entropies, relationships between graph structures, graph energies, topological indices and generalized graph entropies, complexity for calculation of graph entropies. Various applications of graph entropies together with some open problems and conjectures are also presented for further research.