Park, Min Ji; Kim, Eun Sup; Lim, Jung Wook The zero-divisor graph of \(\mathbb{Z}_n[X]]\). (English) Zbl 1462.05180 Kyungpook Math. J. 60, No. 4, 723-729 (2020). Summary: Let \(\mathbb{Z}_n\) be the ring of integers modulo \(n\) and let \(\mathbb{Z}_n[X]]\) be either \(\mathbb{Z}_n [X]\) or \(\mathbb{Z}_n[[X]]\). Let \( \Gamma(\mathbb{Z}_n[X]])\) be the zero-divisor graph of \( \mathbb{Z}_n[X]]\). In this paper, we study some properties of \( \Gamma(\mathbb{Z}_n[X]])\). More precisely, we completely characterize the diameter and the girth of \( \Gamma(\mathbb{Z}_n[X]])\). We also calculate the chromatic number of \( \Gamma(\mathbb{Z}_n[X]])\). Cited in 1 Document MSC: 05C25 Graphs and abstract algebra (groups, rings, fields, etc.) 05C12 Distance in graphs 05C15 Coloring of graphs and hypergraphs 05C38 Paths and cycles 13B25 Polynomials over commutative rings 13F25 Formal power series rings Keywords:\( \Gamma(\mathbb{Z}_n[X]])\); diameter; girth; clique; chromatic number PDFBibTeX XMLCite \textit{M. J. Park} et al., Kyungpook Math. J. 60, No. 4, 723--729 (2020; Zbl 1462.05180) Full Text: DOI References: [1] D. F. Anderson, M. C. Axtell, and J. A. Stickles, Jr,Zero-divisor graphs in commutative rings, Commutative Algebra: Noetherian and Non-Noetherian Perspectives, 23-45, Springer, New York, 2011. · Zbl 1225.13002 [2] D. F. Anderson and P. S. Livingston,The zero-divisor graph of a commutative ring, J. Algebra,217(1999), 434-447. · Zbl 0941.05062 [3] D. D. Anderson and M. Naseer,Beck’s coloring of a commutative ring, J. Algebra, 159(1993), 500-514. · Zbl 0798.05067 [4] M. F. Atiyah and I. G. MacDonald,Introduction to commutative algebra, AddisonWesley Series in Math., Westview Press, 1969. · Zbl 0175.03601 [5] M. Axtell, J. Coykendall, and J. Stickles,Zero-divisor graphs of polynomials and power series over commutative rings, Comm. Algebra,33(2005), 2043-2050. · Zbl 1088.13006 [6] I. Beck,Coloring of commutative rings, J. Algebra,116(1988), 208-226. · Zbl 0654.13001 [7] D. E. Fields,Zero divisors and nilpotent elements in power series rings, Proc. Amer. Math. Soc.,27(1971), 427-433. · Zbl 0219.13023 [8] S. Mulay,Cycles and symmetries of zero-divisors, Comm. Algebra,30(2002), 3533- 3558. · Zbl 1087.13500 [9] S. J. Pi, S. H. Kim, and J. W. Lim,The zero-divisor graph of the ring of integers modulon, Kyungpook Math. J.,59(2019), 591-601. · Zbl 1444.05064 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.