Some results on the co-intersection graph of submodules of a module
Lotf Ali Mahdavi; Yahya Talebi
Commentationes Mathematicae Universitatis Carolinae (2018)
- Volume: 59, Issue: 1, page 15-24
- ISSN: 0010-2628
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topMahdavi, 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- Akbari S., Nikandish R., Nikmehr M. J., 10.1142/S0219498812502003, J. Algebra Appl. 12 (2013), no. 4, 1250200, 13 pp. Zbl1264.05056DOI10.1142/S0219498812502003
- Akbari S., Tavallaee A., Khalashi Ghezelahmad S., 10.1142/S0219498811005452, J. Algebra Appl. 11 (2012), no. 1, 1250019, 8 pp. DOI10.1142/S0219498811005452
- Akbari S., Tavallaee A., Khalashi Ghezelahmad S., 10.1142/S0219498815501169, J. Algebra Appl. 14 (2015), 1550116, 11 pp. DOI10.1142/S0219498815501169
- 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
- Anderson F. W., Fuller K. R., Rings and Categories of Modules, Springer, New York, 1992. Zbl0765.16001
- Bondy J. A., Murty U. S. R., Graph Theory, Graduate Texts in Mathematics, 244, Springer, New York, 2008. Zbl1134.05001
- Bosak J., The graphs of semigroups, in Theory of Graphs and Its Application, Academic Press, New York, 1964, pp. 119–125. Zbl0161.20901
- 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
- Clark J., Lomp C., Vanaja N., Wisbauer R., Lifting Modules. Supplements and Projectivity in Module Theory, Frontiers in Mathematics, Birkhäuser, Basel, 2006.
- Cohn P. M., Introduction to Ring Theory, Springer Undergraduate Mathematics Series, Springer, London, 2000.
- Csakany B., Pollak G., The graph of subgroups of a finite group, Czechoslovak Math. J. 19 (1969), 241–247.
- Jafari S., Jafari Rad N., Planarity of intersection graphs of ideals of rings, Int. Electron. J. Algebra 8 (2010), 161–166.
- 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
- 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
- Mahdavi L. A., Talebi Y., Co-intersection graph of submodules of a module, Algebra Discrete Math. 21 (2016), no. 1, 128–143.
- Northcott D. G., Lessons on Rings, Modules and Multiplicaties, Cambridge University Press, Cambridge, 1968.
- Shen R., 10.1007/s10587-010-0085-4, Czechoslovak Math. J. 60(4) (2010), 945–950. DOI10.1007/s10587-010-0085-4
- Talebi A. A., 10.3844/jmssp.2012.82.84, J. Mathematics Statistics 8 (2012), no. 1, 82–84. DOI10.3844/jmssp.2012.82.84
- Yaraneri E., 10.1142/S0219498812502180, J. Algebra Appl. 12 (2013), no. 5, 1250218, 30 pp. DOI10.1142/S0219498812502180
- Zelinka B., Intersection graphs of finite abelian groups, Czechoslovak Math. J. 25(2) (1975), 171–174.
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