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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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\textit{H. Su} and \textit{Y. Wei}, Taiwanese J. Math. 23, No. 1, 1--10 (2019; Zbl 1406.05044)

### References:

[1] | M. Afkhami and F. Khosh-Ahang, Unit graphs of rings of polynomials and power series, Arab. J. Math. 2 (2013), no. 3, 233–246. · Zbl 1270.05054 |

[2] | S. Akbari, E. Estaji and M. R. Khorsandi, On the unit graph of a non-commutative ring, Algebra Colloq. 22 (2015), Special Issue no. 1, 817–822. · Zbl 1328.05075 |

[3] | D. F. Anderson and A. Badawi, The total graph of a commutative ring, J. Algebra 320 (2008), no. 7, 2706–2719. · Zbl 1158.13001 |

[4] | D. F. Anderson and P. S. Livingston, The zero-divisor graph of a commutative ring, J. Algebra 217 (1999), no. 2, 434–447. · Zbl 0941.05062 |

[5] | D. F. Anderson and S. B. Mulay, On the diameter and girth of a zero-divisor graph, J. Pure Appl. Algebra 210 (2007), no. 2, 543–550. · Zbl 1119.13005 |

[6] | N. Ashrafi, H. R. Maimani, M. R. Pournaki and S. Yassemi, Unit graphs associated with rings, Comm. Algebra 38 (2010), no. 8, 2851–2871. · Zbl 1219.05150 |

[7] | N. Ashrafi and P. Vámos, On the unit sum number of some rings, Q. J. Math. 56 (2005), no. 1, 1–12. · Zbl 1100.11036 |

[8] | I. Beck, Coloring of commutative rings, J. Algebra 116 (1988), no. 1, 208–226. · Zbl 0654.13001 |

[9] | F. R. DeMeyer, T. McKenzie and K. Schneider, The zero-divisor graph of a commutative semigroup, Semigroup Forum 65 (2002), no. 2, 206–214. · Zbl 1011.20056 |

[10] | R. P. Grimaldi, Graphs from rings, Proceedings of the Twentieth Southeastern Conference on Combinatorics, Graph Theory, and Computing (Boca Raton, FL, 1989), Congr. Numer. 71 (1990), 95–103. · Zbl 0747.05091 |

[11] | B. Herwig and M. Ziegler, A remark on sums of units, Arch. Math. (Basel) 79 (2002), no. 6, 430–431. · Zbl 1010.13010 |

[12] | F. Heydari and M. J. Nikmehr, The unit graph of a left Artinian ring, Acta Math. Hungar. 139 (2013), no. 1-2, 134–146. · Zbl 1299.05158 |

[13] | D. Khurana and A. K. Srivastava, Unit sum numbers of right self-injective rings, Bull. Austral. Math. Soc. 75 (2007), no. 3, 355–360. · Zbl 1119.16029 |

[14] | ——–, Right self-injective rings in which every element is a sum of two units, J. Algebra Appl. 6 (2007), no. 2, 281–286. · Zbl 1116.16033 |

[15] | H. R. Maimani, M. R. Pournaki and S. Yassemi, Necessary and sufficient conditions for unit graphs to be Hamiltonian, Pacific J. Math. 249 (2011), no. 2, 419–429. · Zbl 1214.05074 |

[16] | H. Su, K. Noguchi and Y. Zhou, Finite commutative rings with higher genus unit graphs, J. Algebra Appl. 14 (2015), no. 1, 1550002, 14 pp. · Zbl 1325.13010 |

[17] | H. Su, G. Tang and Y. Zhou, Rings whose unit graphs are planar, Publ. Math. Debrecen 86 (2015), no. 3-4, 363–376. · Zbl 1349.05169 |

[18] | H. Su and Y. Zhou, On the girth of the unit graph of a ring, J. Algebra Appl. 13 (2014), no. 2, 1350082, 12 pp. · Zbl 1283.05130 |

[19] | P. Vámos, \(2\)-good rings, Q. J. Math. 56 (2005), no. 3, 417–430. |

[20] | K. G. Wolfson, An ideal-theoretic characterization of the ring of all linear transformations, Amer. J. Math. 75 (1953), 358–386. · Zbl 0050.11503 |

[21] | D. Zelinsky, Every linear transformation is a sum of nonsingular ones, Proc. Amer. Math. Soc. 5 (1954), 627–630. · Zbl 0056.11002 |

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