Homogeneity and self-isometry count of MCD-spaces

Establish that every finite maximal-center-of-distances ultrametric space is homogeneous and that the cardinality of its self-isometry group satisfies \(\log_2|\operatorname{Is}(X)|=|X|-1\).

Background

The paper defines a maximal-center-of-distances (MCD) space as a finite ultrametric space (X,d)(X,d) satisfying ∣X∣=2∣C(X)∣−1|X|=2^{|C(X)|-1}. It then introduces homogeneity in terms of the existence of a self-isometry mapping any point to any other point.

The conjecture proposes two unresolved properties for every MCD-space: homogeneity and an exact formula for the number of self-isometries. The authors explicitly state that they believe this conjecture is true, but do not prove it.

References

The authors believe the following conjecture is true. Let (X,d) be a ${\bf MCD}$-space and let \operatorname{Is}(X) be the set of all self-isometries of (X,d). Then the following statements hold.

— Maximal Center of Distances of Finite Ultrametric Spaces and Perfect Binary Trees  (2609.11384 - Dovgoshey et al., 10 Sep 2026) in Section 5, Conjecture \ref{llp}

Describe the structure of the groups \operatorname{Is}(X) of ${\bf MCD}$-spaces (X,d) up to isomorphism.

— Maximal Center of Distances of Finite Ultrametric Spaces and Perfect Binary Trees  (2609.11384 - Dovgoshey et al., 10 Sep 2026) in Section 5, Problem \ref{gapl}