The radius of unit graphs of rings

Zhiqun Li; Huadong Su; Zhiqun Li; Huadong Su

doi:10.3934/math.2021667

AIMS Mathematics

2021, Volume 6, Issue 10: 11508-11515. doi: 10.3934/math.2021667

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Research article

The radius of unit graphs of rings

Zhiqun Li ,
Huadong Su ^,

School of Sciences, Beibu Gulf University, Qinzhou 535011, China

Received: 28 April 2021 Accepted: 04 August 2021 Published: 09 August 2021
MSC : 16U60, 05C25

Let $ R $ be a ring with nonzero identity. The unit graph of $ R $ is a simple graph whose vertex set is $ R $ itself and two distinct vertices are adjacent if and only if their sum is a unit of $ R $. In this paper, we study the radius of unit graphs of rings. We prove that there exists a ring $ R $ such that the radius of unit graph can be any given positive integer. We also prove that the radius of unit graphs of self-injective rings are 1, 2, 3, $ \infty $. We classify all self-injective rings via the radius of its unit graph. The radius of unit graphs of some ring extensions are also considered.
- unit graph,
- radius,
- self-injective ring,
- unit sum number,
- ring extension
Citation: Zhiqun Li, Huadong Su. The radius of unit graphs of rings[J]. AIMS Mathematics, 2021, 6(10): 11508-11515. doi: 10.3934/math.2021667

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Abstract

Let $ R $ be a ring with nonzero identity. The unit graph of $ R $ is a simple graph whose vertex set is $ R $ itself and two distinct vertices are adjacent if and only if their sum is a unit of $ R $. In this paper, we study the radius of unit graphs of rings. We prove that there exists a ring $ R $ such that the radius of unit graph can be any given positive integer. We also prove that the radius of unit graphs of self-injective rings are 1, 2, 3, $ \infty $. We classify all self-injective rings via the radius of its unit graph. The radius of unit graphs of some ring extensions are also considered.

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