Let $c,ϵ>0$ and $n$ be sufficiently large. If $A\subset \mathbb{N}$ has $|A|=n$ and $G$ is any graph on $A$ with at least $n^{1+c}$ edges then$$\max (|A{+}_{G}A|,|A{\cdot }_{G}A|)\ge |A{|}^{1+c-ϵ},$$where$$A{+}_{G}A=\{a+b:(a,b)\in G\}$$and similarly for $A{\cdot }_{G}A$.