  In this paper, we convert the lattice configurations into networks with
different modes of links and consider models on networks with arbitrary numbers
of interacting particle-pairs. We solve the Heisenberg model by revealing the
relation between the Casimir operator of the unitary group and the
conjugacy-class operator of the permutation group. We generalize the Heisenberg
model by this relation and give a series of exactly solvable models. Moreover,
by numerically calculating the eigenvalue of Heisenberg models and random walks
on network with different numbers of links, we show that a system on lattice
configurations with interactions between more particle-pairs have higher
degeneracy of eigenstates. The highest degeneracy of eigenstates of a lattice
model is discussed.
