  We study the degeneracy of the ground-state energy $E$ of the two-component
Bose-Hubbard model and of the perturbative correction $E_1$. We show that the
degeneracy properties of $E$ and $E_1$ are closely related to the connectivity
properties of the lattice. We determine general conditions under which $E$ is
nondegenerate. This analysis is then extended to investigate the degeneracy of
$E_1$. In this case, in addition to the lattice structure, the degeneracy also
depends on the number of particles present in the system. After identifying the
cases in which $E_1$ is degenerate and observing that the standard (degenerate)
perturbation theory is not applicable, we develop a method to determine the
zeroth-order correction to the ground state by exploiting the symmetry
properties of the lattice. This method is used to implement the perturbative
approach to the two-component Bose-Hubbard model in the case of degenerate
$E_1$ and is expected to be a valid tool to perturbatively study the asymmetric
character of the Mott-insulator to superfluid transition between the particle
and hole side.
