Broadcasting information in a network is an important function in networking applications. In some networks, as wireless sensor networks or some ad-hoc networks it is so essential as to dominate the performance of the entire system. Exploiting some recent results based on the computation of the eigenvector centrality of nodes in the network graph and classical dynamic diffusion models on graphs, this paper derives a novel theoretical framework for efficient information broadcasting in mesh networks with low duty-cycling without the need to build a distribution tree. The model provides lower and upper stochastic bounds with high probability. We show that the lower bound is very close to the theoretical optimum and that a preliminary implementation provides results that are very close to the lower bound on classical graph models.