The diameter of the nil-clean graph of ${\mathbb{Z}}_n$

Let $R$ be a ring with identity. An element $r$ was called to be nil-clean if $r$ was a sum of an idempotent and a nilpotent element in $R$. The nil-clean graph of $R$ was a simple graph, denoted by $G_{NC}(R)$, whose vertex set was $R$, where two distinct vertices $x$ and $y$ were adjacent if, and only if, $x+y$ was a nil-clean element of $R$. In the absence of the condition that vertex $x$ is not the same as $y$, the graph defined in the same way was called the closed nil-clean graph of $R$, which may contain loops, and was denoted by $\overline{G_{NC}}(R)$. In this short note, we completely determine the diameter of $G_{NC}({\mathbb{Z}}_n)$.