On Perfect Co-Annihilating-Ideal Graph of a Commutative Artinian Ring

Download PDF


DOI: 10.46793/KgJMat2101.063M


Let R be a commutative ring with identity. The co-annihilating-ideal graph of R, denoted by AR, is a graph whose vertex set is the set of all non-zero proper ideals of R and two distinct vertices I and J are adjacent whenever Ann(I) Ann(J) = (0). In this paper, we characterize all Artinian rings for which both of the graphs AR and AR (the complement of AR), are chordal. Moreover, all Artinian rings whose AR (and thus AR) is perfect are characterized.


Co-annihilating-ideal graph, perfect graph, chordal graph.


[1]   S. Akbari, A. Alilou, J. Amjadi and S. M. Sheikholeslami, The co-annihilating-ideal graphs of commutative rings, Canad. Math. Bull. 60 (2017), 3–11.

[2]   D. F. Anderson and A. Badawi, On the total graph of a commutative ring without the zero element, J. Algebra Appl. 11 (2012), Article ID 1250074, 18 pages.

[3]   D. F. Anderson and P. S. Livingston, The zero-divisor graph of a commutative ring, J. Algebra 217 (1999), 434–447.

[4]   M. F. Atiyah and I. G. Macdonald, Introduction to Commutative Algebra, Addison-Wesley Publishing Company, Reading, Massachusetts, 1969.

[5]   A. Badawi, On the annihilator graph of a commutative ring, Comm. Algebra 42 (2014), 108–121.

[6]   A. Badawi, On the dot product graph of a commutative ring, Comm. Algebra 43 (2015), 43–50.

[7]   W. Bruns and J. Herzog, Cohen-Macaulay Rings, Cambridge University Press, Cambridge, 1997.

[8]   R. Diestel, Graph Theory, Springer-Verlag, New York, USA, 2000.

[9]   G. A. Dirac, On rigid circuit graphs, Abh. Math. Semin. Univ. Hambg. 38 (1961), 71–76.

[10]   A.  Mallika and R. Kala, A note on zero-divisor graph of amalgamated duplication of a ring along an ideal, AKCE Int. J. Graphs Comb. 14 (2017), 18–26.

[11]   R.  Nikandish, M. J. Nikmehr and M. Bakhtyiari, Coloring of the the annihilator graph of a commutative ring, J. Algebra Appl. 15 (2016), Article ID 1650124, 13 pages.

[12]   T. T. Chelvam and K. Selvakumar, On the connectivity of the annihilating-ideal graphs, Discuss. Math. Gen. Algebra Appl. 35 (2015), 195–204.

[13]   D. B. West, Introduction to Graph Theory, 2nd ed., Prentice Hall, Upper Saddle River, 2001.