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In this paper, we consider the communication problem that involves transmission of correlated sources over broadcast channels. We consider a graph-based framework for this information transmission problem. The system involves a source coding module and a channel coding module. In the source coding module, the sources are efficiently mapped into a nearly semi-regular bipartite graph, and in the channel coding module, the edges of this graph are reliably transmitted over a broadcast channel. We consider nearly semi-regular bipartite graphs as discrete interface between source coding and channel coding in this multiterminal setting. We provide an information-theoretic characterization of 1) the rate of exponential growth (as a function of the number of channel uses) of the size of the bipartite graphs whose edges can be reliably transmitted over a broadcast channel and 2) the rate of exponential growth (as a function of the number of source samples) of the size of the bipartite graphs which can reliably represent a pair of correlated sources to be transmitted over a broadcast channel.