where the total sum in the first term does not depend on σ σ , imply that minimizing H ( σ σ ) in σ σ is equivalent to minimizing ∑ ∑ i j ∈ ∈ δ δ ( V + ) J i j . Defining the edge weight W i j = − − J i j thus turns the Ising problem without an external field into a graph Max-Cut problem maximizing the cut size | δ δ ( V + ) | , which is related to the Ising Hamiltonian as follows,