  Motivated by the problem of zero-error broadcasting, we introduce a new
notion of graph capacity, termed $\rho$-capacity, that generalizes the Shannon
capacity of a graph. We derive upper and lower bounds on the $\rho$-capacity of
arbitrary graphs, and provide a Lov\'asz-type upper bound for regular graphs.
We study the behavior of the $\rho$-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 $\rho$-capacity.
