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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 isomorphism conjecture for caterpillars, an extensively studied problem.

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