A co-ideal based identity-summand graph of a commutative semiring

S. Ebrahimi Atani; S. Dolati Pish Hesari; M. Khoramdel

Commentationes Mathematicae Universitatis Carolinae (2015)

  • Volume: 56, Issue: 3, page 269-285
  • ISSN: 0010-2628

Abstract

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Let I be a strong co-ideal of a commutative semiring R with identity. Let Γ I ( R ) be a graph with the set of vertices S I ( R ) = { x ∈ R ∖ I : x + y ∈ I for some y ∈ R ∖ I } , where two distinct vertices x and y are adjacent if and only if x + y ∈ I . We look at the diameter and girth of this graph. Also we discuss when Γ I ( R ) is bipartite. Moreover, studies are done on the planarity, clique, and chromatic number of this graph. Examples illustrating the results are presented.

How to cite

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Atani, S. Ebrahimi, Hesari, S. Dolati Pish, and Khoramdel, M.. "A co-ideal based identity-summand graph of a commutative semiring." Commentationes Mathematicae Universitatis Carolinae 56.3 (2015): 269-285. <http://eudml.org/doc/271564>.

@article{Atani2015,
abstract = {Let $I$ be a strong co-ideal of a commutative semiring $R$ with identity. Let $\Gamma _\{I\} (R)$ be a graph with the set of vertices $S_\{I\} (R) = \lbrace x \in R\setminus I: x + y \in I$ for some $y \in R \setminus I\rbrace $, where two distinct vertices $x$ and $y$ are adjacent if and only if $x + y \in I$. We look at the diameter and girth of this graph. Also we discuss when $\Gamma _\{I\} (R)$ is bipartite. Moreover, studies are done on the planarity, clique, and chromatic number of this graph. Examples illustrating the results are presented.},
author = {Atani, S. Ebrahimi, Hesari, S. Dolati Pish, Khoramdel, M.},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {strong co-ideal; $Q$-strong co-ideal; identity-summand element; identity-summand graph; co-ideal based},
language = {eng},
number = {3},
pages = {269-285},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {A co-ideal based identity-summand graph of a commutative semiring},
url = {http://eudml.org/doc/271564},
volume = {56},
year = {2015},
}

TY - JOUR
AU - Atani, S. Ebrahimi
AU - Hesari, S. Dolati Pish
AU - Khoramdel, M.
TI - A co-ideal based identity-summand graph of a commutative semiring
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2015
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 56
IS - 3
SP - 269
EP - 285
AB - Let $I$ be a strong co-ideal of a commutative semiring $R$ with identity. Let $\Gamma _{I} (R)$ be a graph with the set of vertices $S_{I} (R) = \lbrace x \in R\setminus I: x + y \in I$ for some $y \in R \setminus I\rbrace $, where two distinct vertices $x$ and $y$ are adjacent if and only if $x + y \in I$. We look at the diameter and girth of this graph. Also we discuss when $\Gamma _{I} (R)$ is bipartite. Moreover, studies are done on the planarity, clique, and chromatic number of this graph. Examples illustrating the results are presented.
LA - eng
KW - strong co-ideal; $Q$-strong co-ideal; identity-summand element; identity-summand graph; co-ideal based
UR - http://eudml.org/doc/271564
ER -

References

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