### Connectivity of The Triple Idempotent Graph of Ring Zn

Vika Yugi Kurniawan, Bayu Purboutomo, Sutrima Sutrima, Nughthoh Arfawi Kurdhi

#### Abstract

Let R be a commutative ring and I(R) denotes a set of all idempotent elements of R. The triple idempotent graph of ring R, denoted by T I(R), is the undirected simple graph with vertex-set in R−{0,1}. Two distinct vertices u and v in T I(Zn) are adjacent if and only if there exists w ∈ R− {0,1} where w ̸= u and w ̸= v such as uv ∈/ I(R), uw ∈/ I(R), vw ∈/ I(R) and uvw ∈ I(R). In this research, we study the connectivity of the triple idempotent graph of ring integer modulo n, denoted by T I(Zn). The result is that the triple idempotent graph of ring Zn is a connected graph if n prime and n ≥ 7.

#### Keywords

the triple idempotent graph; ring of integers modulo n; connected graph

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#### References

Akbari, S., Kiani, D., Mohammadi, F. and Moradi, S.: The total graph

and regular graph of a commutative ring. Journal of pure and applied

algebra, 213(12), pp.2224-2228 (2019).

Anderson, D. F., Livingston, P.S.:The Zero-Divisor Graph of a Commutative Ring. J. Algebra, 217,pp: 434-447 (1999).

Akhtar, R. and Lee, L.: Connectivity of the zero-divisor graph for finite rings. Involve, a Journal of Mathematics, 9(3), pp.415-422 (2016).

Beck, I.: Coloring of Commutative Ring. J. Algebra, Vol. 116, No. 1, pp: 208-226 (1988).

C¸ elikel, E.C.: Triple Zero Graph of Commutative Ring. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, Vol. 70 (2), pp: 653-663 (2021).

Chartrand, G.: Introductory Graphs Theory. Dover Publicaion. Inc, New York (1997).

Chartrand, G., Lesniak L. and Zhang, P.: Graphs & Digraphs. Discrete

Applied Mathematics. Dover Publicaion. Inc, New York (2012).

Mariiaa, H., Kurniawan, V.Y. and Sutrima. : Zero annihilator graph of

semiring of matrices over Boolean semiring. AIP Conference roceedings. Vol. 2326. No. 1 (2021).

Mohammad, H.Q. dan Shuker, N.H. : The Idempotent Divisor Graph of Commutative Ring. Iraqi Journal, Vol. 63, No. 2, pp: 645-651 (2022).

Osba, E.A., Al-Addasi, S. and Jaradeh, N.A.: Zero divisor graph for the ring of Gaussian integers modulo n. Communications in Algebra, 36(10), pp.3865-3877 (2008).

Rahmawati, A., Kurniawan, V.Y. and Wibowo, S.: The diameter of

annihilator ideal graph of Zn. AIP Conference Proceedings. Vol. 2326.

No. 1, pp.020020 (2021).

DOI: http://dx.doi.org/10.12962/j24775401.v10i1.20266

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