Czechoslovak Mathematical Journal, Vol. 67, No. 1, pp. 151-169, 2017

# Some results on the annihilator graph of a commutative ring

## Mojgan Afkhami, Kazem Khashyarmanesh, Zohreh Rajabi

#### Received August 13, 2015.   First published February 24, 2017.

Mojgan Afkhami, Department of Mathematics, University of Neyshabur, P. O. Box 91136-899, Neyshabur, Iran, e-mail: mojgan.afkhami@yahoo.com; Kazem Khashyarmanesh, Zohreh Rajabi, Department of Pure Mathematics, Ferdowsi University of Mashhad, P. O. Box 1159-91775, Mashhad, Iran, e-mail: Khashyar@ipm.ir, rajabi261@yahoo.com

Abstract: Let $R$ be a commutative ring. The annihilator graph of $R$, denoted by ${\rm AG}(R)$, is the undirected graph with all nonzero zero-divisors of $R$ as vertex set, and two distinct vertices $x$ and $y$ are adjacent if and only if ${\rm ann}_R(xy) \neq{\rm ann}_R(x)\cup{\rm ann}_R(y)$, where for $z \in R$, ${\rm ann}_R(z) = \lbrace r \in R \colon rz = 0\rbrace$. In this paper, we characterize all finite commutative rings $R$ with planar or outerplanar or ring-graph annihilator graphs. We characterize all finite commutative rings $R$ whose annihilator graphs have clique number $1$, $2$ or $3$. Also, we investigate some properties of the annihilator graph under the extension of $R$ to polynomial rings and rings of fractions. For instance, we show that the graphs ${\rm AG}(R)$ and ${\rm AG}(T(R))$ are isomorphic, where $T(R)$ is the total quotient ring of $R$. Moreover, we investigate some properties of the annihilator graph of the ring of integers modulo $n$, where $n \geq1$.

Keywords: annihilator graph; zero-divisor graph; outerplanar; ring-graph; cut-vertex; clique number; weakly perfect; chromatic number; polynomial ring; ring of fractions

Classification (MSC 2010): 05C75, 13A99, 05C99

DOI: 10.21136/CMJ.2017.0436-15

Full text available as PDF.

References:
[1] M. Afkhami: When the comaximal and zero-divisor graphs are ring graphs and outerplanar. Rocky Mt. J. Math. 44 (2014), 1745-1761. DOI 10.1216/RMJ-2014-44-6-1745 | MR 3310946 | Zbl 1306.05092
[2] M. Afkhami, Z. Barati, K. Khashyarmanesh: When the unit, unitary and total graphs are ring graphs and outerplanar. Rocky Mt. J. Math. 44 (2014), 705-716. DOI 10.1216/RMJ-2014-44-3-705 | MR 3264477 | Zbl 1301.05075
[3] S. Akbari, H. R. Maimani, S. Yassemi: When a zero-divisor graph is planar or a complete {$r$}-partite graph. J. Algebra 270 (2003), 169-180. DOI 10.1016/S0021-8693(03)00370-3 | MR 2016655 | Zbl 1032.13014
[4] D. F. Anderson, M. C. Axtell, J. A. Stickles, Jr.: Zero-divisor graphs in commutative rings. Commutative Algebra. Noetherian and Non-Noetherian Perspectives M. Fontana et al. Springer, New York (2011), 23-45. DOI 10.1007/978-1-4419-6990-3_2 | MR 2762487 | Zbl 1225.13002
[5] D. F. Anderson, A. Badawi: On the zero-divisor graph of a ring. Commun. Algebra 36 (2008), 3073-3092. DOI 10.1080/00927870802110888 | MR 2440301 | Zbl 1152.13001
[6] D. F. Anderson, A. Badawi: The total graph of a commutative ring. J. Algebra 320 (2008), 2706-2719. DOI 10.1016/j.jalgebra.2008.06.028 | MR 2441996 | Zbl 1158.13001
[7] D. F. Anderson, R. Levy, J. Shapiro: Zero-divisor graphs, von Neumann regular rings, and Boolean algebras. J. Pure Appl. Algebra 180 (2003), 221-241. DOI 10.1016/S0022-4049(02)00250-5 | MR 1966657 | Zbl 1076.13001
[8] D. F. Anderson, P. S. Livingston: The zero-divisor graph of a commutative ring. J. Algebra 217 (1999), 434-447. DOI 10.1006/jabr.1998.7840 | MR 1700509 | Zbl 0941.05062
[9] D. D. Anderson, M. Naseer: Beck's coloring of a commutative ring. J. Algebra 159 (1993), 500-514. DOI 10.1006/jabr.1993.1171 | MR 1231228 | Zbl 0798.05067
[10] N. Ashrafi, H. R. Maimani, M. R. Pournaki, S. Yassemi: Unit graphs associated with rings. Commun. Algebra 38 (2010), 2851-2871. DOI 10.1080/00927870903095574 | MR 2730284 | Zbl 1219.05150
[11] M. F. Atiyah, I. G. Macdonald: Introduction to Commutative Algebra. Series in Mathematics, Addison-Wesley Publishing Company, Reading, London (1969). MR 0242802 | Zbl 0175.03601
[12] A. Badawi: On the annihilator graph of a commutative ring. Commun. Algebra 42 (2014), 108-121. DOI 10.1080/00927872.2012.707262 | MR 3169557 | Zbl 1295.13006
[13] A. Badawi: On the dot product graph of a commutative ring. Commun. Algebra 43 (2015), 43-50. DOI 10.1080/00927872.2014.897188 | MR 3240402 | Zbl 1316.13005
[14] Z. Barati, K. Khashyarmanesh, F. Mohammadi, K. Nafar: On the associated graphs to a commutative ring. J. Algebra Appl. 11 (2012), 1250037, 17 pages. DOI 10.1142/S0219498811005610 | MR 2925450 | Zbl 1238.13015
[15] I. Beck: Coloring of commutative rings. J. Algebra 116 (1988), 208-226. DOI 10.1016/0021-8693(88)90202-5 | MR 0944156 | Zbl 0654.13001
[16] R. Belshoff, J. Chapman: Planar zero-divisor graphs. J. Algebra 316 (2007), 471-480. DOI 10.1016/j.jalgebra.2007.01.049 | MR 2354873 | Zbl 1129.13028
[17] B. CotÃ©, C. Ewing, M. Huhn, C. M. Plaut, D. Weber: Cut-sets in zero-divisor graphs of finite commutative rings. Commun. Algebra 39 (2011), 2849-2861. DOI 10.1080/00927872.2010.489534 | MR 2834134 | Zbl 1228.13011
[18] I. Gitler, E. Reyes, R. H. Villarreal: Ring graphs and complete intersection toric ideals. Discrete Math. 310 (2010), 430-441. DOI 10.1016/j.disc.2009.03.020 | MR 2564795 | Zbl 1198.05089
[19] A. Kelarev: Graph Algebras and Automata. Pure and Applied Mathematics 257, Marcel Dekker, New York (2003). MR 2064147 | Zbl 1070.68097
[20] A. Kelarev: Labelled Cayley graphs and minimal automata. Australas. J. Comb. 30 (2004), 95-101. MR 2080457 | Zbl 1152.68482
[21] A. Kelarev, J. Ryan, J. Yearwood: Cayley graphs as classifiers for data mining: The influence of asymmetries. Discrete Math. 309 (2009), 5360-5369. DOI 10.1016/j.disc.2008.11.030 | MR 2548552 | Zbl 1206.05050
[22] H. R. Maimani, M. Salimi, A. Sattari, S. Yassemi: Comaximal graph of commutative rings. J. Algebra 319 (2008), 1801-1808. DOI 10.1016/j.jalgebra.2007.02.003 | MR 2383067 | Zbl 1141.13008
[23] D. B. West: Introduction to Graph Theory. Prentice Hall, Upper Saddle River (1996). MR 1367739 | Zbl 0845.05001