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Let    G  =  (  V  ,  E  )    be a graph and    σ  (  G  )    the number of independent (vertex) sets of  G . Then the Merrifield–Simmons conjecture states that the sign of the term    σ  (  G  −  u  )  ·  σ  (  G  −  v  )  −  σ  (  G  )  ·  σ  (  G  −  u  −  v  )    only depends on the parity of the distance of the vertices    u  ,  v  ∈  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.

The Merrifield–Simmons Conjecture Holds for Bipartite Graphs