The bridge-free matroid criterion for isomorphism of graph algebras

Let $G_1$ and $G_2$ be graphs, and let their bridge-free matroids be the graphical matroids of their respective $\Delta$-subgraphs.

Bridge-free matroid criterion. The algebras $\mathcal C_{G_1}^T$ and $\mathcal C_{G_2}^T$ are isomorphic if and only if their bridge-free matroids are isomorphic.

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