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]).