Comparing phylogenetic trees of stars with spectral graph distances

Phylogenetic trees offer a hierarchical representation of chemical similarities among stars, from which their evolutionary histories can, in principle, be reconstructed. Simulation-based work suggests that phylogenetic trees built under different chemical evolution scenarios carry distinguishable structural imprints, but whether existing methods can detect this signal in practice is unclear. We evaluated a range of distance measures -- both classical tree-based and graph-theoretical, in terms of their ability to compare such trees. Our goal was to identify measures that are robust to observational noise and that separate trees built from populations with distinct chemical enrichment histories. We constructed phylogenetic trees using the Neighbor-Joining algorithm from chemical abundance vectors of solar twins and of stars associated with the Milky Way (MW) disk, Gaia–Sausage–Enceladus (GSE), and the Sagittarius dwarf spheroidal galaxy (Sgr). We then compared trees using classical tree-based and graph-based measures. Two benchmark tests assess: (1) stability under input perturbations within the same evolutionary history and (2) the discriminative power between distinct stellar populations used as proxies for different evolutionary histories. Classical tree metrics rapidly saturate under perturbations of order the typical abundance uncertainty (approx2σ, or approx0.05 dex per element) and fail to distinguish trees when leaf sets differ. In contrast, spectral distances vary smoothly with perturbation amplitude and provide a label-invariant framework for comparing trees built from disjoint sets of stars, including populations with distinct chemical evolutionary histories. Among the spectral measures evaluated, the Laplacian spectral distance shows the most consistent separation across the population pairs, while the normalized variant is the weakest due to its reduced sensitivity to branch lengths. Applied to MW, GSE, and Sgr stars, our spectral measures recover a distance ordering consistent with the known chemical enrichment histories of these populations. Beyond distinguishing well-characterized populations, this framework offers a means to compare, for example, an observed enrichment history with simulated ones to identify which best reproduces the observed tree structure.