  Bonnet, Kim, Thomass\'{e}, and Watrigant (2020) introduced the twin-width of
a graph. We show that the twin-width of an $n$-vertex graph is less than
$(n+\sqrt{n\ln n}+\sqrt{n}+2\ln n)/2$, and the twin-width of an $m$-edge graph
for a positive $m$ is less than $\sqrt{3m}+ m^{1/4} \sqrt{\ln m} / (4\cdot
3^{1/4}) + 3m^{1/4} / 2$. Conference graphs of order $n$ (when such graphs
exist) have twin-width at least $(n-1)/2$, and we show that Paley graphs
achieve this lower bound. We also show that the twin-width of the
Erd\H{o}s-R\'{e}nyi random graph $G(n,p)$ with $1/n\leq p=p(n)\leq 1/2$ is
larger than $2p(1-p)n - (2\sqrt{2}+\varepsilon)\sqrt{p(1-p)n\ln n}$
asymptotically almost surely for any positive $\varepsilon$. Lastly, we
calculate the twin-width of random graphs $G(n,p)$ with $p\leq c/n$ for a
constant $c<1$, determining the thresholds at which the twin-width jumps from
$0$ to $1$ and from $1$ to $2$.
