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.