Isomorphism of weighted trees and Stanley's isomorphism conjecture for caterpillars

This paper contributes to a programme initiated by the first author: "How much information about a graph is revealed in its Potts partition function?“ We show that the W -polynomial distinguishes non-isomorphic weighted trees of a good family. The framework developed to do so also allows us to show that the W -polynomial distinguishes non-isomorphic caterpillars. This establishes Stanley's conjecture for caterpillars, an extensively studied problem.