The diameter of unit graphs of rings.(English)Zbl 1406.05044

Summary: 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 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 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 [F. Heydari and M. J. Nikmehr, Acta Math. Hung. 139, No. 1–2, 134–146 (2013; Zbl 1299.05158), Theorem 2 and Corollary 1].

MSC:

 05C25 Graphs and abstract algebra (groups, rings, fields, etc.) 16U60 Units, groups of units (associative rings and algebras) 05C12 Distance in graphs

Zbl 1299.05158
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References:

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