1 Introduction  
2 Inequalities and extremal properties on (generalized) graph entropies  
2.1 Inequalities for classical graph entropies and parametric measures  
2.2 Graph entropy inequalities with information functions  
2.3 Information theoretic measures of UHG graphs  
2.4 Bounds for the entropies of rooted trees and generalized trees  
2.6 Extremal properties of degree-based and distance-based graph entropies  
2.7 Extremality of certain graph entropies and entropy bounds for dendrimers  
2.8 Sphere-regular graphs and extremality entropies  
2.9 Information inequalities for generalized graph entropies  
3 Relationships between graph structures, graph energies, topological indices and generalized graph entropies  
4 Summary and conclusion