Random d-regular graphs have sum-sets of size n^{1-2/d} for every abelian group, proving a polynomial lower bound that is tight up to polylog factors.
Bollob´ as,Random Graphs, second edition, Cambridge University Press, 2001
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Sums along the edges of bounded degree graphs
Random d-regular graphs have sum-sets of size n^{1-2/d} for every abelian group, proving a polynomial lower bound that is tight up to polylog factors.