In 1982, B. Bollobas and Vega in the paper gave the configurational model to generate $r$-regular random graphs. They gave the following theorem (Theorem 1 in the paper).

Theorem:Let $r\geq 3$ and $\epsilon> 0$ be fixed and define $d=d(n)$ as the least integer satisfying $$(r-1)^{d-1}\geq (2+\epsilon)rn \ \text{log} \ n.$$ Then almost every $r$-regular graphs of order $n$ has diameter at most $d$.

I have following two questions

Question 1.Does this theorem implies that the diameter of $r$-regular random graphs is of order $\text{log} (n \text{log} n),$ that is, $d(n)=O(\text{log} (n \text{log} n))$?

Question 2.If Question 1 is true, will all the $r$-regular random graphs (not necessary generated using the configuration model) has the diameter of order $\text{log} (n \text{log} n).$