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

Abstract

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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 Ω ( M ) , is an undirected simple graph whose vertex set V ( Ω ) is a set of all nontrivial submodules of M and two distinct vertices N and K are adjacent if and only if N + K ≠ M . We study the connectivity, the core and the clique number of Ω ( M ) . Also, we provide some conditions on the module M , under which the clique number of Ω ( M ) is infinite and Ω ( M ) is a planar graph. Moreover, we give several examples for which n the graph Ω ( ℤ n ) is connected, bipartite and planar.

How to cite

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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 -

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