Evans et al. [1] proved the subadditivity of the mutual information in the broadcasting on tree model with binary vertex labels and symmetric edge channels. They raised the question of whether such subadditivity extends to loopy graphs in some appropriate way. We propose here such a generalization for general graphs and binary vertex labels. With enough channel symmetry, the generalization applies to arbitrary graphs, and with partial symmetry, it applies to series-parallel graphs. The results are obtained using the Chi-squared mutual information rather than the classical KL-mutual information (for which some of our bounds do not hold). Various properties of the Chi-squared mutual information are discussed.