  The Lieb-Mattis theorem about antiferromagnetic ordering of energy levels on
bipartite lattices is generalized to finite-size two-leg spin-1/2 ladder model
frustrated by diagonal interactions. For reflection-symmetric model with
site-dependent interactions we prove exactly that the lowest energies in
sectors with fixed total spin and reflection quantum numbers are monotone
increasing functions of total spin. The nondegeneracy of most levels is proved
also. We also establish the uniqueness and obtain the spin value of the
lowest-level multiplet in the whole sector formed by reflection-symmetric
(antisymmetric) states. For a wide range of coupling constants, we prove that
the ground state is a unique spin singlet. For other values of couplings, it
may be also a unique spin triplet or may consist of both multiplets. Similar
results have been obtained for the ladder with arbitrary boundary impurity
spin. Some partial results have also been obtained in the case of periodical
boundary conditions.
