# Article

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Keywords:
strong co-ideal; $Q$-strong co-ideal; identity-summand element; identity-summand graph; co-ideal based
Summary:
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 $S_{I} (R) = \{x \in R\setminus I: x + y \in I$ for some $y \in R \setminus I\}$, 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.
References:
[1] Anderson D.F., Livingston P.S.: The zero-divisor graph of a commutative ring. J. Algebra 217 (1999), 434–447. DOI 10.1006/jabr.1998.7840 | MR 1700509 | Zbl 1035.13004
[2] Akbari S., Maimani H.R., Yassemi S.: When a zero-divisor graph is planar or a complete r-partite graph. J. Algebra 270 (2003), 169–180. DOI 10.1016/S0021-8693(03)00370-3 | MR 2016655 | Zbl 1032.13014
[3] Beck I.: Coloring of commutative rings. J. Algebra 116 (1988), 208–226. DOI 10.1016/0021-8693(88)90202-5 | MR 0944156 | Zbl 0654.13001
[4] Bondy J. A., Murty U.S.R.: Graph Theory. Graduate Texts in Mathematics, 244, Springer, New York, 2008. MR 2368647 | Zbl 1134.05001
[5] Ebrahimi Atani S.: The zero-divisor graph with respect to ideals of a commutative semiring. Glas. Mat. 43(63) (2008), 309–320. DOI 10.3336/gm.43.2.06 | MR 2460702 | Zbl 1162.16031
[6] Ebrahimi Atani S.: An ideal-based zero-divisor graph of a commutative semiring. Glas. Mat. 44(64) (2009), 141–153. DOI 10.3336/gm.44.1.07 | MR 2525659 | Zbl 1181.16041
[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. MR 3041341
[8] Ebrahimi Atani S., Dolati Pish Hesari S., Khoramdel M.: The identity-summand graph of commutative semirings. J. Korean Math. Soc. 51 (2014), 189-202. DOI 10.4134/JKMS.2014.51.1.189 | MR 3159324
[9] Dolžan D., Oblak P.: The zero-divisor graphs of rings and semirings. Internat. J. Algebra Comput. 22 (2012), no. 4, 1250033, 20 pp. DOI 10.1142/S0218196712500336 | MR 2946298 | Zbl 1251.05075
[10] Golan J.S.: Semirings and Their Applications. Kluwer Academic Publishers, Dordrecht, 1999. MR 1746739 | Zbl 0947.16034
[11] Maimani H.R., Pournaki M.R, Yassemi S.: Zero-divisor graph with respect to an ideal. Comm. Algebra 34 (2006), 923–929. DOI 10.1080/00927870500441858 | MR 2208109 | Zbl 1092.13004
[12] Redmond P.: An ideal-based zero-divisor graph of a commutative ring. Comm. Algebra 31 (2003), 4425–4443. DOI 10.1081/AGB-120022801 | MR 1995544 | Zbl 1020.13001
[13] Spiroff S., Wickham C.: A zero divisor graph determined by equivalence classes of zero divisors. Comm. Algebra 39 (2011), 2338–2348. DOI 10.1080/00927872.2010.488675 | MR 2821714 | Zbl 1225.13007
[14] Wang H.: On rational series and rational language. Theoret. Comput. Sci. 205 (1998), 329–336. DOI 10.1016/S0304-3975(98)00103-0 | MR 1638617

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