Diameter and radius of an additive-multiplicative graph

Let $G$ be the graph on the positive integers in which distinct vertices are adjacent if they differ by one or their quotient in one order is prime. We prove that $6\le \mathrm{diam}(G)\le 7$ and $4\le \mathrm{rad}(G)\le 5$. Under Dickson’s conjecture for two linear forms, we prove that $\mathrm{diam}(G)=6$ and $\mathrm{rad}(G)=4$.