Connectivity of The Triple Idempotent Graph of Ring Zn

Vika Yugi Kurniawan (Universitas Sebelas Maret)
Bayu Purboutomo (Universitas Sebelas Maret)
Sutrima Sutrima (Universitas Sebelas Maret)
Nughthoh Arfawi Kurdhi (Universitas Sebelas Maret)

International Journal of Computing Science and Applied Mathematics
Vol 10, No 1 (2024)

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.