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February, 2019 The Diameter of Unit Graphs of Rings
Taiwanese J. Math. 23(1): 1-10 (February, 2019). DOI: 10.11650/tjm/180602

## Abstract

Let $R$ be a ring. The unit graph of $R$, denoted by $G(R)$, is the simple graph defined on all elements of $R$, and where two distinct vertices $x$ and $y$ are linked by an edge if and only if $x+y$ is a unit of $R$. The diameter of a simple graph $G$, denoted by $\operatorname{diam}(G)$, is the longest distance between all pairs of vertices of the graph $G$. In the present paper, we prove that for each integer $n \geq 1$, there exists a ring $R$ such that $n \leq \operatorname{diam}(G(R)) \leq 2n$. We also show that $\operatorname{diam}(G(R)) \in \{ 1,2,3,\infty \}$ for a ring $R$ with $R/J(R)$ self-injective and classify all those rings with $\operatorname{diam}(G(R)) = 1,2,3$ and $\infty$, respectively. This extends [12, Theorem 2 and Corollary 1].

## Citation

Huadong Su. Yangjiang Wei. "The Diameter of Unit Graphs of Rings." Taiwanese J. Math. 23 (1) 1 - 10, February, 2019. https://doi.org/10.11650/tjm/180602

## Information

Received: 31 August 2017; Revised: 12 May 2018; Accepted: 4 June 2018; Published: February, 2019
First available in Project Euclid: 15 June 2018

zbMATH: 07021715
MathSciNet: MR3909987
Digital Object Identifier: 10.11650/tjm/180602

Subjects:
Primary: 05C25 , 16U60
Secondary: 05C12

Keywords: $k$-good , diameter , self-injective ring , Unit graph , unit sum number

Rights: Copyright © 2019 The Mathematical Society of the Republic of China

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