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 $Gamma(R)$, is the graph whose vertices are the non-zero  zero-divisors of  $R$, and two distinct vertices $r$ and $s$ 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 $0leq mbox{diam}(Gamma(R))leq mbox{diam}(Gamma(R[M]))leq 3$. Moreover, we give a characterization for the possible diam$(Gamma(R))$ and diam$(Gamma(R[M]))$, when $R$ is a reversible Noetherian ring and $M$ is a u.p.-monoid. Also, we study relations between the girth of $Gamma(R)$ and that of $Gamma(R[M])$.