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→Definition: The Ising Hamiltonian is an example of a pseudo-Boolean function |
→Connection to graph maximum cut: indent |
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For the Ising model without an external field on a graph G, the Hamiltonian becomes the following sum over the graph edges E(G)
:<math>H(\sigma) = -\sum_{ij\in E(G)} J_{ij}\sigma_i\sigma_j</math>.
Here each vertex i of the graph is a spin site that takes a spin value <math>\sigma_i = \pm 1 </math>. A given spin configuration <math>\sigma</math> partitions the set of vertices <math>V(G)</math> into two <math>\sigma</math>-depended subsets, those with spin up <math>V^+</math> and those with spin down <math>V^-</math>. We denote by <math>\delta(V^+)</math> the <math>\sigma</math>-depended set of edges that connects the two complementary vertex subsets <math>V^+</math> and <math>V^-</math>. The ''size'' <math>\left|\delta(V^+)\right|</math> of the cut <math>\delta(V^+)</math> to [[bipartite graph|bipartite]] the weighted undirected graph G can be defined as
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