Extension of the Lieb-Schupp theorem to the Heisenberg models with higher order interactions

We extend the Lieb-Schupp theorem to the Heisenberg models with higher order interactions on non-frustrated or frustrated finite lattices. These lattices are constructed by even numbered rings with or without crossing bonds and have reflection symmetry. The results show that all ground states have total spin zero in wide interaction parameters region which is not covered with the results of the Marshall-Lieb-Mattis type arguments.