We investigate the stability domains of ground states of generalized Hubbard models with next-nearest neighbour interaction using the optimum groundstate approach. We focus on the $eta$-pairing state with momentum P=0 and the fully polarized ferromagnetic state at half-filling. For these states exact lower bounds for the regions of stability are obtained in the form of inequalities between the interaction parameters. For the model with only nearest neighbour interaction we show that the bounds for the stability regions can be improved by considering larger clusters. Additional next-nearest neighbour interactions can lead to larger or smaller stability regions depending on the parameter values.
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