Nonisomorphic trees are not LV-equivalent

Prove that for every integer n ≥ 2, any two nonisomorphic trees on n vertices yield n-component homogeneous Lotka–Volterra tree-systems that are not LV-equivalent, meaning there is no invertible linear transformation of variables that maps one system to the other while preserving the Darboux-polynomial structure (and thus their associated admissible hypergraphs are not equivalent).

Background

The paper studies parametric classes of n-component Lotka–Volterra (LV) systems that admit additional linear Darboux polynomials (DPs) and associates them with loopless admissible hypergraphs whose hyperedges record the variable supports of those DPs. Linear changes of variables can transform one LV-system into another while altering the hypergraph structure (because DPs are preserved but their variable supports need not be), inducing an equivalence relation on admissible hypergraphs called LV-equivalence.

Tree-systems are a well-studied subclass in which the additional linear DPs involve pairs of variables corresponding to edges of a tree on n vertices; these systems are maximally superintegrable. The authors classified admissible hypergraphs and LV-equivalence classes for n ≤ 5, and computed, for n < 9, the sizes of sets of hypergraphs LV-equivalent to each tree. Motivated by this evidence, they posit the conjecture that nonisomorphic trees always determine non-LV-equivalent LV-systems, i.e., tree-systems are distinguished up to the LV-equivalence induced by linear transformations.