Home Mathematics Chordal and Perfect Zero-Divisor Graphs of Posets and Applications to Graphs Associated with Algebraic Structures
Article
Licensed
Unlicensed Requires Authentication

Chordal and Perfect Zero-Divisor Graphs of Posets and Applications to Graphs Associated with Algebraic Structures

  • Nilesh Khandekar and Vinayak Joshi EMAIL logo
Published/Copyright: October 7, 2023
Become an author with De Gruyter Brill

ABSTRACT

In this paper, we characterize the perfect zero-divisor graphs and chordal zero-divisor graphs (its complement) of ordered sets. These results are applied to the zero-divisor graphs of finite reduced rings, the comaximal ideal graphs of rings, the annihilating ideal graphs of rings, the intersection graphs of ideals of rings, and the intersection graphs of subgroups of groups. In fact, it is shown that these graphs associated with a commutative ring R with identity can be effectively studied via the zero-divisor graph of a specially constructed poset from R.

2020 Mathematics Subject Classification: Primary 05C17; 05C25; 06A07; 13A70

(Communicated by Anatolij Dvurečenskij)


Funding statement: The first author is financially supported by the Council of Scientific and Industrial Research(CSIR), New Delhi, via Senior Research Fellowship Award Letter No. 09/137(0620)/2019-EMR-I.

REFERENCES

[1] Aghapouramin, V.—Nikmehr, M. J.: Perfectness of a graph associated with annihilating ideals of a ring, Discrete Math. Algorithms Appl. 10(4) (2018), Art. ID 1850047.10.1142/S1793830918500477Search in Google Scholar

[2] Afkhami, M.—Barati, Z.—Khashyarmanesh, K.: Planar zero divisor graphs of partially ordered sets, Acta Math. Hungar. 137(1-2) (2012), 27–35.10.1007/s10474-012-0231-6Search in Google Scholar

[3] Akbari, S.—Alilou, A.—Amjadi, J.—Sheikholeslami, S. M.: The co-annihilating-ideal graphs of commutative rings, Canad. Math. Bull. 60(1) (2017), 3–11.10.4153/CMB-2016-017-1Search in Google Scholar

[4] Anderson, D. F.—Lagrange, J. D.: Some remarks on the compressed zero-divisor graph, J. Algebra 447 (2016), 297–321.10.1016/j.jalgebra.2015.08.021Search in Google Scholar

[5] Azadi, M.—Jafari, Z.—Eslahchi, C.: On the comaximal ideal graph of a commutative ring, Turkish J. Math. 40 (2016), 905–913.10.3906/mat-1505-23Search in Google Scholar

[6] Atiyah, M. F.—Macdonald, I. G.: Introduction to Commutative Algebra, Addison-Wesley, Reading, MA, 1969.Search in Google Scholar

[7] Bagheri, S.—Nabael, F.—Rezaeii, R.—Samei, K.: Reduction graph and its application on algebraic graphs, Rocky Mountain J. Math. 48(3) (2018), 729–751.10.1216/RMJ-2018-48-3-729Search in Google Scholar

[8] Beck, I.: Coloring of a commutative ring, J. Algebra 116 (1988), 208–226.10.1016/0021-8693(88)90202-5Search in Google Scholar

[9] Chajda, I.—Eigenthaler, G.—Langer, H.: Ideals of direct products of rings, Asian–Eur. J. Math. 11(4) (2018), Art. ID 1850094.10.1142/S1793557118500948Search in Google Scholar

[10] Chakrabarty, I.—Ghosh, S.—Mukherjee, T. K.—SEN, M. K.: Intersection graphs of ideals of rings, Discrete Math. 309(17) (2009), 5381–5392.10.1016/j.disc.2008.11.034Search in Google Scholar

[11] Chameni-Nembua, C.—Monjardet, B.: Finite pseudocomplemented lattices, European J. Combin. 13(2) (1992), 89–107.10.1016/0195-6698(92)90041-WSearch in Google Scholar

[12] Chudnovsky, M.—Robertson, N.—Seymour, P.—Thomas, R.: The strong perfect graph theorem, Ann. of Math. 164(1) (2006), 51–229.10.4007/annals.2006.164.51Search in Google Scholar

[13] Das, A.: On perfectness of intersection graph of ideals of Zn, Discuss. Math. Gen. Algebra Appl. 37(2) (2017), 119–126.10.7151/dmgaa.1270Search in Google Scholar

[14] Devhare, S.—Joshi, V.—Lagrange, J. D.: Eulerian and Hamiltonian complements of zero-divisor graphs of pseudocomplemented posets, Palest. J. Math. 8(2) (2019), 30–39.Search in Google Scholar

[15] Dirac, G. A.: On rigid circuit graphs, Abh. Math. Semin. Univ. Hambg. 38 (1961), 18–26.Search in Google Scholar

[16] Hungerford, T. W.: On the structure of principal ideal rings, Pacific J. Math. 25(3) (1968), 543–547.10.2140/pjm.1968.25.543Search in Google Scholar

[17] Janowitz, M. F.: Section semicomplemented lattices, Math. Z. 63 (1968), 63–76.10.1007/BF01110457Search in Google Scholar

[18] Halaš, R.—Jukl, M.: On Beck’s coloring of partially ordered sets, Discrete Math. 309 (2009), 4584–4589.10.1016/j.disc.2009.02.024Search in Google Scholar

[19] Halaš, R.: Pseudocomplemented ordered sets, Arch. Math. (Brno) 29 (1993), 153–160.Search in Google Scholar

[20] Halaš, R.: Some properties of Boolean ordered sets, Czechoslovak Math. J. 46 (1996), 93–98.10.21136/CMJ.1996.127273Search in Google Scholar

[21] Joshi, V.: On completion of section semicomplemented posets, Southeast Asian Bull. Math. 31 (2007), 881–892.Search in Google Scholar

[22] Joshi, V.: Zero divisor graph of a poset with respect to an ideal, Order 29 (2012), 499–506.10.1007/s11083-011-9216-2Search in Google Scholar

[23] Joshi, V.—Khiste, A.: The zero-divisor graphs of Boolean posets, Math. Slovaca 64 (2014), 511–519.10.2478/s12175-014-0221-ySearch in Google Scholar

[24] Joshi, V.—Mundlik, N.: Baer ideals in 0-distributive posets, Asian–Eur. J. Math. 9(3) (2016), Art. ID 1650055.10.1142/S1793557116500558Search in Google Scholar

[25] Joshi, V.—Pourali, H. Y.—Waphare, B. N.: The graph of equivalence classes of zero divisors, Int. Sch. Res. Notices (2014), Art. ID 896270.10.1155/2014/896270Search in Google Scholar

[26] Joshi, V.—Waphare, B. N.: Characterization of 0-distributive posets, Math. Bohem. 130(1) (2005), 73–80.10.21136/MB.2005.134222Search in Google Scholar

[27] Karp, R. M.: Reducibility among combinatorial problems. In: Complexity of Computer Computations (Miller R. E., Thatcher J. W., Bohlinger J. D., eds.), The IBM Research Symposia Series, Springer, Boston, MA, 1972, pp. 85–103.10.1007/978-1-4684-2001-2_9Search in Google Scholar

[28] Khandekar, N.—Joshi, V.: Zero-divisor graphs and total coloring conjecture, Soft Comput. 24 (2020), 18273–18285.10.1007/s00500-020-05344-2Search in Google Scholar

[29] Lagrange, J. D.—ROY, K. A.: Poset graphs and the lattice of graph annihilators, Discrete Math. 313(10) (2013), 1053–1062.10.1016/j.disc.2013.02.004Search in Google Scholar

[30] Larmerová, J.—RachŬnek, J.: Translations of distributive and modular ordered sets, Acta Univ. Palacki. Olomuc., Fac. Rerum Nat., Math. 91 (1988), 13–23.Search in Google Scholar

[31] Lovász, L.: A characterization of perfect graphs, J. Combin. Theory Ser. B 13(2) (1972), 95–98.10.1016/0095-8956(72)90045-7Search in Google Scholar

[32] Lu, D.—WU, T.: The zero-divisor graphs of partially ordered sets and an application to semigroups, Graphs Combin. 26 (2010), 793–804.10.1007/s00373-010-0955-4Search in Google Scholar

[33] Mirghadim, S. M.—Nikmehr, M. J.—Nikandish, R.: On perfect co-annihilating-ideal graph of a commutative Artinian ring, Kragujevac J. Math. 45(1) (2021), 63–73.10.46793/KgJMat2101.063MSearch in Google Scholar

[34] Nikseresht, A.: Chordality of graphs associated to commutative rings, Turkish J. Math. 42 (2018), 2202–2213.10.3906/mat-1801-71Search in Google Scholar

[35] Patil, A.—Waphare, B. N.—Joshi, V.: Perfect zero-divisor graphs, Discrete Math. 340(4) (2017), 740–745.10.1016/j.disc.2016.11.027Search in Google Scholar

[36] Smith, B.: Perfect zero-divisor graphs of n , Rose-Hulman Undergrad. Math J. 17(2) (2016), 111–132.Search in Google Scholar

[37] Stern, M.: Semimodular Lattices Theory and Applications, Cambridge University Press, 1999.10.1017/CBO9780511665578Search in Google Scholar

[38] Venkatanarasimhan, P. V.: Pseudo-complements in posets, Proc. Amer. Math. Soc. 28(1) (1971), 9–17.10.1090/S0002-9939-1971-0272687-XSearch in Google Scholar

[39] Visweswaran, S.—Patel, H. D.: A graph associated with the set of all nonzero annihilating ideals of a commutative ring, Discrete Math. Algorithms Appl. 6(4) (2014), Art. ID 1450047.10.1142/S1793830914500475Search in Google Scholar

[40] Ye, M.—Wu, T. S.: Comaximal ideal graphs of commutative rings, J. Algebra Appl. 11(6) (2012), Art. ID 1250114.10.1142/S0219498812501149Search in Google Scholar

[41] Waphare, B. N.—Joshi, V.: On uniquely complemented posets, Order 22 (2005), 11–20.10.1007/s11083-005-9002-0Search in Google Scholar

[42] West, D. B.: Introduction to Graph Theory, 2nd ed., Prentice Hall, 2001.Search in Google Scholar

Received: 2022-08-22
Accepted: 2023-01-05
Published Online: 2023-10-07

© 2023 Mathematical Institute Slovak Academy of Sciences

Downloaded on 15.12.2025 from https://www.degruyterbrill.com/document/doi/10.1515/ms-2023-0081/pdf
Scroll to top button