With respect to the notation in , we are taking here $N=n+1$.

We say that $u$ is an $s$-minimal graph in $\Omega_0$ if $E_u$ is a locally $s$-minimal set in $\Omega_0\times{\ensuremath{\mathbb{R}}}$.

Interestingly, $s$-minimal graphs enjoy suitable Bernstein-type properties, see [@MR3680376; @PISA; @2018arXiv180705774C], and they have a smooth interior regularity theory, as proved in [@MR3934589].

See also [@NOCHETTO] for several very precise simulations on nonlocal minimal graphs and a sharp numerical analysis of their properties.

With this, we are ready to state the main result of this article, which gives some precise geometric conditions on the trace of nonlocal minimal graphs. We establish that the trace graph has necessarily [*zero derivatives when the trace crosses zero*]{}. The precise result that we have is the following one:

\[NLMS\] Let $u$ be an $s$-minimal graph in $(-2,2)\times(0,4)$. Assume that there exists $h>0$ such that $u=0$ in $ (-2,2)\times(-h,0)$, and let $$\label{MR3516886EQ}
f(x_1):=\lim_{x_2\searrow0}u(x_1,x_2).$$ Then, there exist $\delta_0\in\left(0,\frac1{100}\right)$ and $C>0$ such that if $\zeta_0\in\left(-\frac{3}2,\frac32\right)$ is such that $f(\zeta_0)=0$, then $$\label{VANP} |u(x)|{\leqslant}C\,|x-(\zeta_0,0)|^{\frac{3+s}2}$$ for every $x\in B_{\delta_0}(\zeta_0,0)$, and, in particular, $$\label{THAN-2}
{\mbox{$f'(\zeta_0)=0$.}}$$

We remark that the existence of the limit in  is warranted by Theorem 1.1 in [@MR3516886].

![*[[A nonlocal minimal graph, in the light of Theorem 