Motivated by the problem of zero-error broadcasting, we introduce a new notion of graph capacity, termed $ρ$-capacity, that generalizes the Shannon capacity of a graph. We derive upper and lower bounds on the $ρ$-capacity of arbitrary graphs, and provide a Lovász-type upper bound for regular graphs. We study the behavior of the $ρ$-capacity under two graph operations: the strong product and the disjoint union. Finally, we investigate the connection between the structure of a graph and its $ρ$-capacity.