Connectivity of The Triple Idempotent Graph of Ring Zn

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.