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 Lovasz-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.
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