Proper $\{a,b\}$-edge-weightings of trees

What the math gives to ML

The paper gives a constructive binary edge labeling for trees such that adjacent vertices have different weighted incident-edge sums. This can transfer to tree-structured GNNs as a deterministic symmetry-breaking positional encoding, avoiding random node identifiers and adding no trainable parameters. The generic case with nonzero distinct weights and nonzero sum is the most useful regime because every tree except K2 admits a valid labeling. The zero-sum and zero-weight cases are specialized degeneracies and are less attractive for a first implementation.