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

top
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

top

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

top
  1. Anderson D.F., Livingston P.S., 10.1006/jabr.1998.7840, J. Algebra 217 (1999), 434–447. Zbl1035.13004MR1700509DOI10.1006/jabr.1998.7840
  2. Akbari S., Maimani H.R., Yassemi S., 10.1016/S0021-8693(03)00370-3, J. Algebra 270 (2003), 169–180. Zbl1032.13014MR2016655DOI10.1016/S0021-8693(03)00370-3
  3. Beck I., 10.1016/0021-8693(88)90202-5, J. Algebra 116 (1988), 208–226. Zbl0654.13001MR0944156DOI10.1016/0021-8693(88)90202-5
  4. Bondy J. A., Murty U.S.R., Graph Theory, Graduate Texts in Mathematics, 244, Springer, New York, 2008. Zbl1134.05001MR2368647
  5. Ebrahimi Atani S., 10.3336/gm.43.2.06, Glas. Mat. 43(63) (2008), 309–320. Zbl1162.16031MR2460702DOI10.3336/gm.43.2.06
  6. Ebrahimi Atani S., 10.3336/gm.44.1.07, Glas. Mat. 44(64) (2009), 141–153. Zbl1181.16041MR2525659DOI10.3336/gm.44.1.07
  7. Ebrahimi Atani S., Dolati Pish Hesari S., Khoramdel M., Strong co-ideal theory in quotients of semirings, J. Adv. Res. Pure Math. 5(3) (2013), 19–32. MR3041341
  8. Ebrahimi Atani S., Dolati Pish Hesari S., Khoramdel M., 10.4134/JKMS.2014.51.1.189, J. Korean Math. Soc. 51 (2014), 189-202. MR3159324DOI10.4134/JKMS.2014.51.1.189
  9. Dolžan D., Oblak P., 10.1142/S0218196712500336, Internat. J. Algebra Comput. 22 (2012), no. 4, 1250033, 20 pp. Zbl1251.05075MR2946298DOI10.1142/S0218196712500336
  10. Golan J.S., Semirings and Their Applications, Kluwer Academic Publishers, Dordrecht, 1999. Zbl0947.16034MR1746739
  11. Maimani H.R., Pournaki M.R, Yassemi S., 10.1080/00927870500441858, Comm. Algebra 34 (2006), 923–929. Zbl1092.13004MR2208109DOI10.1080/00927870500441858
  12. Redmond P., 10.1081/AGB-120022801, Comm. Algebra 31 (2003), 4425–4443. Zbl1020.13001MR1995544DOI10.1081/AGB-120022801
  13. Spiroff S., Wickham C., 10.1080/00927872.2010.488675, Comm. Algebra 39 (2011), 2338–2348. Zbl1225.13007MR2821714DOI10.1080/00927872.2010.488675
  14. Wang H., 10.1016/S0304-3975(98)00103-0, Theoret. Comput. Sci. 205 (1998), 329–336. MR1638617DOI10.1016/S0304-3975(98)00103-0

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.