Commutative rings introduce a class of identifiable graphs

Document Type : Research Paper

Authors

Department of Mathematics, Jundi-Shapur University of Technology Dezful, Iran

10.22108/toc.2024.138005.2081

Abstract

Let $R$ be a commutative ring with identity, and $ \mathrm{A}(R) $ be the set of ideals with non-zero annihilator. The annihilating-ideal graph of $ R $ is defined as the graph $AG(R)$ with the vertex set $ \mathrm{A}(R)^{*}=\mathrm{A}(R)\setminus\lbrace 0\rbrace $ and two distinct vertices $ I $ and $ J $ are adjacent if and only if $ IJ=0 $. In this paper, we characterize all positive integers $n$ for which $AG(\mathbb{Z}_n)$ is identifiable.

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Main Subjects


[1] G. Aalipour, S. Akbari, R. Nikandish, M. J. Nikmehr and F. Shaveisi, On the coloring of the annihilatingideal graph of a commutative ring, Discrete Math., 312 no. 17 (2012) 2620–2626.
[2] M. F. Atiyah and I. G. Macdonald, Introduction to commutative algebra, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1969.
[3] D. Auger, Identifying codes in trees and planar graphs, European Conference on Combinatorics, Graph Theory and Applications (EuroComb 2009), 585–588, Electron. Notes Discrete Math., Elsevier Sci. B. V.,
Amsterdam, 34 (2009).
[4] C. Balbuena, F. Foucaud and A. Hansberg, Locating dominating sets and identifying codes in graphs of girth at least 5, Electron. J. Combin., 22 no. 2 (2015) 22 pp.
[5] M. Behboodi and Z. Rakeei, The annihilating-ideal graph of commutative rings I, J. Algebra Appl., 10 no. 4 (2011) 727–739.
[6] C. Berge, Graphes, Gauthier-Villars, Paris, 1983.
[7] N. Bertrand, I. Charon, O. Hudry and A. Lobestein, Identifying and locating-dominating codes on chains and cycles, European J. Combin., 25 no. 7 (2004) 969–987.
[8] C. Camarero, C. Martinez and R. Beivide, Identifying codes of degree 4 Cayley graphs over Abelian groups, Adv. Math. Commun., 9 no. 2 (2015) 129–148.
[9] I. Charon, I. Honkala, O. Hudry and A. Lobestein, Minimum sizes of identifying codes in graphs differing by one vertex, Cryptogr. Commun., 5 no. 2 (2013) 119–136.
[10] I. Charon, I. Honkala, O. Hudry and A. Lobestein, Minimum sizes of identifying codes in graphs differing by one edge, Cryptogr. Commun, 6 no. 2 (2014) 157–170.
[11] I. Charon, O. Hudry and A. Lobstein, Extremal cardinalities for identifying and locating-dominating codes in graphs, Discrete Math., 307 no. 3-5 (2007) 356–366.
[12] J. Hedetniemi, On identifying codes in the Cartesian product of a path and a complete graph, J. Comb. Optim., 31 no. 4 (2016) 1405–1416.
[13] M. G. Karpovsky, K. Chakrabarty and L. B. Levitin, On a new class of codes for identifying vertices in graphs, IEEE Trans. Inform. Theory, 44 no. 2 (1998) 599–611.
[14] M. Murtaza, I. Javaid and M. Fazil, Covering codes of a graph associated with a finite vector space, Ukrainian Math. J., 72 no. 7 (2020) 1108–1117.
[15] J. Moncel, On graphs on n vertices having an identifying code of cardinality ⌈log2 (n + 1)⌉, Discrete Appl. Math., 154 no. 14 (2006) 2032–2039.
[16] R. Nikandish, H. R. Maimani and H. Izanloo, The annihilating-ideal graph of Zn is weakly perfect, Contrib. Discrete Math., 11 no. 1 (2016) 16–21.
[17] D. B. West, Introduction to graph theory, Prentice Hall, Inc., Upper Saddle River, NJ, 1996.
[18] H. Y. Yu and T. Wu, Commutative rings R whose C(AG(R)) consist of only triangles, Comm. Algebra, 43 no. 3 (2015) 1076–1097.

Articles in Press, Corrected Proof
Available Online from 21 February 2024
  • Receive Date: 10 June 2023
  • Revise Date: 28 January 2024
  • Accept Date: 05 February 2024
  • Published Online: 21 February 2024