Origin of number of bounds in ising model

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what is the origin of the number of bound (N z /2 ) in the calculating of average in ensemble in Mean Field for the Ising model?

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Each atom has z neigbours and hence z bonds to its neighbours. But each bond is shared by two atoms, so you have Nz/2 bonds in total.
 
Thanks, I want to show that potts model hamiltonian is equal to this J∑si.sj that siis
{(1,0) , (-1/2 , √3/2) , (-1/2, √3/2)} . how should I do it? what is the first step?
 

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