Study of graphs for some weakly – two – involution clean rings

The notion of a weakly 2 – invo – clean has been introduced in this paper to extend the concept of 2 – invo – clean and that of a weakly invo – clean. A ring R is called a weakly 2 – invo – clean if for every element x ∈ R can be written x = u + v + e or x = u + v – e, for some u, v ∈ invo(R) and e∈Id(R). The first basic properties of a weakly=– 2 – invo – clean rings have been considered, including some examples needed in the process. We also provided a condition for the W2ic – element to be an Wic – element. The relationship between strongly weakly N – clean and SW2ic was also studied, with some propositions presented concerning this relationship. The relationship between invo – regular and W2ic was studied. After that, the concept of the graph W2invoCl(Zpn) was introduced, from these graphs we give the number of vertices, edges, and some other properties at Zpn and Zpn,qm and Zpn,qm,hr where all p, q and h are prime numbers except the prime number 2, and n, m and r are positive integer numbers.