On the extremal values of general degree-based graph entropies

In this note we prove one part of the conjecture about upper and lower bounds of the degree-based graph entropy Ik(T) in the class of trees introduced in [S. Cao, M. Dehmer, Y. Shi, Extremality of degree-based graph entropies, Information Sciences 278 (2014) 22-33.] using Lagrange multipliers and Jensen's inequality, and disprove the other part by providing a family of counter-examples. Our main result is the following: the path Pn is unique tree on n vertices that maximizes Ik(T) for k > 0, and the star Sn is unique tree on n vertices that minimizes Ik(T) for k ≥ 1.