Kraus–Dehmer–Schaumann's highly connected minimal-entropy conjecture

Let $G$ be a graph on $n$ vertices, and let $I_{f^\sigma}(G)$ denote the entropy associated with the sphere function. A graph is highly connected here when it is obtained from the complete graph $K_n$ by removing a small number of edges.

Kraus–Dehmer–Schaumann's conjecture. A minimal graph for $I_{f^\sigma}(G)$ is highly connected. In particular, a minimal graph on $n$ vertices has at least $m\geq n/2$ vertices of degree $n-1$.

The conjecture is motivated by computations for graphs on $8$ and $9$ vertices and by the observed structure of minimal graphs. It predicts that extremal graphs for this entropy are close to complete and contain many universal vertices.