The existence of a unique positive critical flux distribution and of a corresponding single positive eigenvalue (k-effective), which is greater than the absolute value of any other eigenvalue, is established for the discrete form of the steady-state multigroup diffusion equations. The assumptions here are considerably less restrictive than in formerly published papers. For example, arbitrary scattering matrices, general fission transfer matrices (not necessarily in multiplicative form), and internal nondiffusion regions are allowed. Furthermore, the transitivity assumption of the problem is replaced by weak conditions of connectedness, which are not only sufficient but also necessary for the existence statements. The theoretical and computational significance of the existence and positivity theorems are discussed. Several examples illustrate the generality of the results and the importance of the conditions of connectedness.