Cao–Dehmer–Schaumann's extremal degree-based entropy conjecture for trees

Let $T$ be a tree with $n$ vertices, and let $l>0$. Here $P_n$ and $S_n$ denote the path and star on $n$ vertices, respectively, and $I_{f^l_d}$ is the degree-based graph entropy.

Cao–Dehmer–Schaumann's conjecture. We have

$$I_{f^l_d}(T)\leq I_{f^l_d}(P_n),$$

with equality if and only if $T\cong P_n$, and

$$I_{f^l_d}(T)\geq I_{f^l_d}(S_n),$$

with equality if and only if $T\cong S_n$.

The conjecture was proposed on the basis of numerical experiments after several unsuccessful attempts at a proof. It asserts that, for every positive exponent, paths and stars give the maximum and minimum degree-based entropies among trees, respectively.