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

Abstract

<p>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) $.</p>