The Merrifield-Simmons conjecture holds for bipartite graphs

Let $G = (V, E)$ be a graph and $σ(G)$ the number of independent (vertex) sets in $G$. Then the Merrifield-Simmons conjecture states that the sign of the term $σ(G_{-u}) \cdot σ(G_{-v}) - σ(G) \cdot σ(G_{-u-v})$ only depends on the parity of the distance of the vertices $u, v \in V$ in $G$. We prove that the conjecture holds for bipartite graphs by considering a generalization of the term, where vertex subsets instead of vertices are deleted.

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