Dehmer–Kraus's extremal dendrimer entropy conjecture

Let $D$ be a dendrimer on $n$ vertices with radius $r$ and progressive degree $p$. Let $\bar c_0,\bar c_1,\ldots,\bar c_r$ be a nonincreasing sequence of orbit weights, and let $I_{f^3}(D)$ be the corresponding entropy. The star has $r=1$ and $p=n-2$, while the path has $r=\lceil (n-1)/2\rceil$ and $p=1$.

Dehmer–Kraus's conjecture. For every sequence

$$\bar c_0\geq \bar c_1\geq\cdots\geq\bar c_r,$$

the star graph has the maximal value and the path graph has the minimal value of $I_{f^3}(D)$.

The conjecture was proposed from numerical experiments, with ideas toward a proof. It concerns the extremal entropy among dendrimers for all nonincreasing orbit-weight sequences.