High-ordered spectral characterization conjecture for trees

Let $T$ and $T'$ be trees. Their high-ordered spectrum is the collection of high-ordered spectral invariants associated with each tree.

High-ordered spectral characterization conjecture. Two trees are isomorphic if and only if they have the same high-ordered spectrum.

This conjecture proposes that high-ordered spectral data completely determines a tree up to isomorphism. The paper establishes related high-ordered spectral invariants and notes that no non-isomorphic high-ordered cospectral graphs have been found, leaving the existence of such graphs open.