Conjectured degree of compatibility for multi-entropy graphs

Prove that for the multi-entropy colored graphs ME_n^D, the degree of compatibility satisfies Δ(ME_n^D) = (n−1) n^{D−2} ((D−1)(D−2)/4) for all integers D ≥ 3 and n > 1.

Background

The paper analyzes distinguishing power of various trace-invariant families and introduces the degree of compatibility Δ(G) as a key large-N combinatorial quantity. For the multi-entropy family ME_nD, the authors propose a closed-form formula for Δ(ME_nD) based on the D=3, n=2 case and supporting numerical evidence.

Establishing this conjecture would provide a precise large-N handle on the typical value of the corresponding generalized Rényi entropies and strengthen the connection between combinatorics of colored graphs and multipartite entanglement diagnostics.

References

Based on the study of ME_23 in Refs.~[...], and supported by numerical analysis, we conjecture $$\Delta(ME_nD) = (n-1)n{D-2}\frac{(D-1)(D-2)}{4} \,.$$

— Tensor invariants for multipartite entanglement classification  (2604.02269 - Carrozza et al., 2 Apr 2026) in Section 5.2, Combinatorial quantities and trace-invariants from the literature (footnote in multi-entropy table)