# Some results on the co-intersection graph of submodules of a module

• Volume: 59, Issue: 1, page 15-24
• ISSN: 0010-2628

top

## Abstract

top
Let $R$ be a ring with identity and $M$ be a unitary left $R$-module. The co-intersection graph of proper submodules of $M$, denoted by $\Omega \left(M\right)$, is an undirected simple graph whose vertex set $V\left(\Omega \right)$ is a set of all nontrivial submodules of $M$ and two distinct vertices $N$ and $K$ are adjacent if and only if $N+K\ne M$. We study the connectivity, the core and the clique number of $\Omega \left(M\right)$. Also, we provide some conditions on the module $M$, under which the clique number of $\Omega \left(M\right)$ is infinite and $\Omega \left(M\right)$ is a planar graph. Moreover, we give several examples for which $n$ the graph $\Omega \left({ℤ}_{n}\right)$ is connected, bipartite and planar.

## How to cite

top

Mahdavi, Lotf Ali, and Talebi, Yahya. "Some results on the co-intersection graph of submodules of a module." Commentationes Mathematicae Universitatis Carolinae 59.1 (2018): 15-24. <http://eudml.org/doc/294567>.

@article{Mahdavi2018,
abstract = {Let $R$ be a ring with identity and $M$ be a unitary left $R$-module. The co-intersection graph of proper submodules of $M$, denoted by $\Omega (M)$, is an undirected simple graph whose vertex set $V(\Omega )$ is a set of all nontrivial submodules of $M$ and two distinct vertices $N$ and $K$ are adjacent if and only if $N+K\ne M$. We study the connectivity, the core and the clique number of $\Omega (M)$. Also, we provide some conditions on the module $M$, under which the clique number of $\Omega (M)$ is infinite and $\Omega (M)$ is a planar graph. Moreover, we give several examples for which $n$ the graph $\Omega (\mathbb \{Z\}_\{n\})$ is connected, bipartite and planar.},
author = {Mahdavi, Lotf Ali, Talebi, Yahya},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {co-intersection graph; core; clique number; planarity},
language = {eng},
number = {1},
pages = {15-24},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Some results on the co-intersection graph of submodules of a module},
url = {http://eudml.org/doc/294567},
volume = {59},
year = {2018},
}

TY - JOUR
AU - Mahdavi, Lotf Ali
AU - Talebi, Yahya
TI - Some results on the co-intersection graph of submodules of a module
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2018
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 59
IS - 1
SP - 15
EP - 24
AB - Let $R$ be a ring with identity and $M$ be a unitary left $R$-module. The co-intersection graph of proper submodules of $M$, denoted by $\Omega (M)$, is an undirected simple graph whose vertex set $V(\Omega )$ is a set of all nontrivial submodules of $M$ and two distinct vertices $N$ and $K$ are adjacent if and only if $N+K\ne M$. We study the connectivity, the core and the clique number of $\Omega (M)$. Also, we provide some conditions on the module $M$, under which the clique number of $\Omega (M)$ is infinite and $\Omega (M)$ is a planar graph. Moreover, we give several examples for which $n$ the graph $\Omega (\mathbb {Z}_{n})$ is connected, bipartite and planar.
LA - eng
KW - co-intersection graph; core; clique number; planarity
UR - http://eudml.org/doc/294567
ER -

## References

top
1. Akbari S., Nikandish R., Nikmehr M. J., 10.1142/S0219498812502003, J. Algebra Appl. 12 (2013), no. 4, 1250200, 13 pp. Zbl1264.05056DOI10.1142/S0219498812502003
2. Akbari S., Tavallaee A., Khalashi Ghezelahmad S., 10.1142/S0219498811005452, J. Algebra Appl. 11 (2012), no. 1, 1250019, 8 pp. DOI10.1142/S0219498811005452
3. Akbari S., Tavallaee A., Khalashi Ghezelahmad S., 10.1142/S0219498815501169, J. Algebra Appl. 14 (2015), 1550116, 11 pp. DOI10.1142/S0219498815501169
4. Akbari S., Tavallaee A., Khalashi Ghezelahmad S., 10.1515/ms-2016-0267, Math. Slovaca 67 (2017), no. 2, 297–304. DOI10.1515/ms-2016-0267
5. Anderson F. W., Fuller K. R., Rings and Categories of Modules, Springer, New York, 1992. Zbl0765.16001
6. Bondy J. A., Murty U. S. R., Graph Theory, Graduate Texts in Mathematics, 244, Springer, New York, 2008. Zbl1134.05001
7. Bosak J., The graphs of semigroups, in Theory of Graphs and Its Application, Academic Press, New York, 1964, pp. 119–125. Zbl0161.20901
8. Chakrabarty I., Gosh S., Mukherjee T. K., Sen M. K., 10.1016/j.disc.2008.11.034, Discrete Math. 309 (2009), 5381–5392. DOI10.1016/j.disc.2008.11.034
9. Clark J., Lomp C., Vanaja N., Wisbauer R., Lifting Modules. Supplements and Projectivity in Module Theory, Frontiers in Mathematics, Birkhäuser, Basel, 2006.
10. Cohn P. M., Introduction to Ring Theory, Springer Undergraduate Mathematics Series, Springer, London, 2000.
11. Csakany B., Pollak G., The graph of subgroups of a finite group, Czechoslovak Math. J. 19 (1969), 241–247.
12. Jafari S., Jafari Rad N., Planarity of intersection graphs of ideals of rings, Int. Electron. J. Algebra 8 (2010), 161–166.
13. Kayacan S., Yaraneri E., 10.4134/JKMS.2015.52.1.081, J. Korean Math. Soc. 52 (2015), no. 1, 81–96. DOI10.4134/JKMS.2015.52.1.081
14. Laison J. D., Qing Y., 10.1016/j.disc.2010.06.042, Discrete Math. 310 (2010), 3413–3416. DOI10.1016/j.disc.2010.06.042
15. Mahdavi L. A., Talebi Y., Co-intersection graph of submodules of a module, Algebra Discrete Math. 21 (2016), no. 1, 128–143.
16. Northcott D. G., Lessons on Rings, Modules and Multiplicaties, Cambridge University Press, Cambridge, 1968.
17. Shen R., 10.1007/s10587-010-0085-4, Czechoslovak Math. J. 60(4) (2010), 945–950. DOI10.1007/s10587-010-0085-4
18. Talebi A. A., 10.3844/jmssp.2012.82.84, J. Mathematics Statistics 8 (2012), no. 1, 82–84. DOI10.3844/jmssp.2012.82.84
19. Yaraneri E., 10.1142/S0219498812502180, J. Algebra Appl. 12 (2013), no. 5, 1250218, 30 pp. DOI10.1142/S0219498812502180
20. Zelinka B., Intersection graphs of finite abelian groups, Czechoslovak Math. J. 25(2) (1975), 171–174.

## NotesEmbed?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.