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The Merrifield-Simmons conjectures states a relation between the distance ofvertices in a simple graph $G$ and the number of independent sets, denoted as$\sigma(G)$, in vertex-deleted subgraphs. Namely, that the sign of the term$\sigma(G_{-u}) \cdot \sigma(G_{-v}) - \sigma(G) \cdot \sigma(G_{-u-v})$ onlydepends on the parity of the distance of $u$ and $v$ in $G$. We prove thisstatement in the case of parity graphs and give some evidence that this resultmay not be further generalized to other classes of graphs.

The Merrifield-Simmons conjecture also holds for parity graphs