On zero divisor graph of unique product monoid rings over Noetherian reversible ring

Let R be an associative ring with identity and Z∗(R) be its set of non-zero zero divisors. The zero-divisor graph of R, denoted by Γ(R), is the graph whose vertices are the non-zero zero-divisors of R, and two distinct vertices rr and ss are adjacent if and only if rs=0 or sr=0. In this paper, we bring some results about undirected zero-divisor graph of a monoid ring over reversible right (or left) Noetherian ring R. We essentially classify the diameter-structure of this graph and show that 0≤diam(Γ(R))≤diam(Γ(R[M]))≤3. Moreover, we give a characterization for the possible diam(Γ(R)) and diam(Γ(R[M])), when R is a reversible Noetherian ring and MM is a u.p.-monoid. Also, we study relations between the girth of Γ(R) and that of Γ(R[M]).