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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

Citations:

Zbl 1299.05158
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References:

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