Pseudo-Von Neumann Regular Graph of Commutative Ring

Let R be a commutative ring. The Von.-Neuman regular (Shortly Vn. Neum. reg.) graph of $\mathop{\text{R}}\limits_{\_}$ is a graph which its vertices are all items of R s. t. thither is an edge between vertices a, b if a+b is a Vn.-Neum. reg. item of R. Here a new definition of the Vn. Neum. reg. graph of R called pseudo–Vn.–Neum. reg. graph of R denoted by P-VG(R) is a graph with all items of R represents a vertex, and two different vertices a, b ∈ R are adjacent iff a = a²b or b = b²a. In this work, the main features of P-VG(R) are studied and some outstanding results. Also, we n_—3 prove if P-VG(R), R= Z_n and n ≥ 3, n is a prime then it is graph has $\frac{\text{n}-3}{2}$ of cycle C₃. Finally, we show that If R=Z_p, where p is a prime number then the PG(R) ⊂ P-VG(R).