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

Huadong Su; Zhunti Liang; Huadong Su; Zhunti Liang

doi:10.3934/math.20241210

AIMS Mathematics

2024, Volume 9, Issue 9: 24854-24859. doi: 10.3934/math.20241210

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Research article

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

Huadong Su ^{1
,
,},
Zhunti Liang ²

1.
School of Science, Beibu Gulf University, Qinzhou 535011, China
2.
School of Mathematics and Information Science, Guangxi University, Nanning 530001, China

Received: 14 June 2024 Revised: 30 July 2024 Accepted: 19 August 2024 Published: 23 August 2024
MSC : 05C25, 13A99

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) $.
- nil-clean graph,
- integers modulo $ n $,
- diameter,
- Chinese remainder theorem
Citation: Huadong Su, Zhunti Liang. The diameter of the nil-clean graph of $ \mathbb{Z}_n $[J]. AIMS Mathematics, 2024, 9(9): 24854-24859. doi: 10.3934/math.20241210

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Abstract

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) $.

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