Expansions of the Potts model partition function along deletions and contractions

Abstract: We establish two expansions of the Potts model partition function of a graph. One is along the deletions of a graph, a rewritten formula given in Biggs (1977). The other is along the contractions of a graph. Then, we specialize the partition function to the chromatic or flow polynomial by the M\"obius inversion formula, and prove two known equations of the two polynomials. One expresses the chromatic polynomial as a weighted sum of flow polynomials of deletions, the other expresses the flow polynomial as a weighted sum of chromatic polynomials of contractions. The proof of the former by Biggs formula is due to Bychkov et al. (2021). The two expressions are considered to be dual in the sense of their forms, and transfer to each other with plane duality. This relation also holds in our expansions of the Potts model partition function. We clarify this duality by using matroid duality. Partition functions can be extended to matroids, and the two expansions can also be extended. The two expansions transfer to each other with matroid duality, so in addition to an elementary combinatorial proof of the two, we give another proof by the "duality" relation between them.