DM-GAA

ISSN 1509-9415 (print version)

ISSN 2084-0373 (electronic version)

https://doi.org/10.7151/dmgaa

Discussiones Mathematicae - General Algebra and Applications

Cite Score (2023): 0.6

SJR (2023): 0.214

SNIP (2023): 0.604

Index Copernicus (2022): 121.02

H-Index: 5

Discussiones Mathematicae - General Algebra and Applications

Article in press

Authors:

P. Pal

Pavel Pal

Department of Mathematics
Bankura University
Bankura-722155, India

email: ju.pavel86@gmail.com

0009-0009-0539-6191

J. Jana

Jyotirmoy Jana

Department of Mathematics
Jadavpur University
Kolkata-700032, India

email: jyotirmoyjana.1996@gmail.com

0009-0008-1659-4639

Title:

On Intersection Graph of Ideals of a Near-ring in terms of its Essential Ideals

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Source:

Discussiones Mathematicae - General Algebra and Applications

Received: 2023-10-11 , Revised: 2024-11-23 , Accepted: 2024-11-24 , Available online: 2025-08-12 , https://doi.org/10.7151/dmgaa.1484

Abstract:

Let N be a near-ring and I(N) denote the set of all non-trivial (i.e., non-zero proper) ideals of N. In this paper, the intersection graph of ideals of N has been introduced as a simple undirected graph denoted by G(N) whose vertex set is I(N) and two vertices I and J are adjacent if and only if I ≠ J and I ∩ J ≠ {0}. The graph G(N) has been studied mainly using the direct sum decomposition and the essential ideals of N. Here, some neces- sary and sufficient conditions for G(N) to be connected and complete have been obtained. Under some restrictions on N and G(N), some graph parameters such as chromatic number, independence number, domination number, hull number, geodetic number as well as some graphical properties of G(N) such as being chordal, star etc. have been obtained.

Keywords:

intersection graph, near-ring, direct summand, essential ideal

References:

  1. S. Akbari, F. Heydari and M. Maghasedi, The intersection graph of a group, J. Algebra Appl. 14(5) 2015.
    https://doi.org/10.1142/S0219498815500656
  2. S. Akbari, R. Nikandish and M.J. Nikmehr, Some results on the intersection graphs of ideals of rings, J. Algebra Appl. 12(4) 2013.
    https://doi.org/10.1142/S0219498812502003
  3. F.G. Ball, D.J. Sirl and P.S. Trapman, Epidemics on random intersection graphs, Ann. Appl. Probab. 24(3) (2014) 1081–1128.
    https://doi.org/10.1214/13-AAP942
  4. S. Bhavanari, S.P. Kuncham and B.S. Kedukodi, Graph of a nearring with respect to an ideal, Commun. Algebra 38(5) (2010) 1957–1967.
    https://doi.org/10.1080/00927870903069645
  5. M. Bloznelis, J. Jaworski and K. Rybarczyk, Component evolution in a secure wireless sensor network, Networks 53(1) (2009) 19-26.
    https://doi.org/10.1002/net.20256
  6. J.A. Bondy, and U.S.R. Murty, Graduate Texts in Mathematics, Graph Theory (Springer, 2008).
  7. J. Bos$\acute{a}$k, The graphs of semigroups, Theory of Graphs and Application (M. Fielder, ed.), (Academic Press, New York, 1964) 119–125.
  8. I. Chakrabarty, S. Ghosh, T.K. Mukherjee and M.K. Sen. Intersection graphs of ideals of rings, Discrete Math. 309(17) (2009) 5381–5392.
    https://doi.org/10.1016/j.disc.2008.11.034
  9. I. Chakrabarty and J. V. Kureethara, A survey on the intersection graphs of ideals of rings, Commun. Comb. Optim. 7(2) (2022) 121–167.
    https://doi.org/10.22049/cco.2021.26990.1176
  10. B. Csákány and G. Pollák, The graph of subgroups of a finite group, Czech. Math. J. 19(2) (1969) 241–247.
    https://doi.org/10.21136/CMJ.1969.100891
  11. A. Das, On perfectness of intersection graph of ideals of $\mathbb{Z}_n$, Discuss. Math. Gen. Algebra Appl. 37(2) (2017) 119–126. arXiv:1611.01153v1
  12. D.S. Malik, M.K. Sen and S. Ghosh, Introduction to Graph Theory (Cengage Learning, Singapore, 2014).
  13. G. Pilz, Near-Rings, Revised edition (North-Holland, Amsterdam, 1983).
  14. N.J. Rad, S.H. Jafari and S. Ghosh, On the intersection graphs of ideals of direct product of rings, Discuss. Math. Gen. Algebra Appl. 34(2) (2014) 191–201.
    https://doi.org/10.7151/dmgaa.1224
  15. Y.V. Reddy and B. Satyanarayana, A Note on N-Groups, Indian J. Pure Appl. Math. 19(9) (1988) 842–845.
  16. E. Szpilrajn-Marczewski, Sur deux proprieéteś des classes d'ensembles $($in french$)$, Fund. Math. 33(1) (1945) 303–307.
  17. E. Szpilrajn-Marczewski, Sur deux propriétés des classes d'ensembles, Fund. Math. 33(1) (1945) 303–307 (Trans. Bronwyn Burlingham. ``A Translation of Sur deux propriétés des classes d'ensembles", University of Alberta, 2009).
  18. D.B. West, Introduction to Graph Theory (Prentice Hall of India, New Delhi, 2003).
  19. B. Zelinka, Intersection graphs of finite abelian groups, Czech. Math. J. 25(2) (1975) 171–174.
  20. J. Zhao, O. Ya$\breve{g}$an and V. Gligor, Secure k-connectivity in wireless sensor networks under an on/off channel model, 2013, IEEE International Symposium on Information Theory (2013) 2790–2794.
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