Undirected Zero-Divisor Graphs and Unique Product Monoid Rings

Let R be an associative ring with identity and Z^*(K)be its set of non-zero zero-divisors.The undirected zero-divisor graph of R、denoted byΓ(R),is the graph whose vert ices are the non-zero zero-divisors of R、and where two distinct verticesγand s are adjacent if and only ifγs=0 or sγ=0.The dist ance bet ween vertices a and b is the length of the shortest path connecting them,and the diameter of the graph,diam(Γ(R)),is the superimum of these distances.In this paper,first we prove some results aboutΓ(R)of a semi-commutative ring R.Then,for a reversible ring R and a unique product monoid M、we prove 0≦diam(Γ(R))<diam(Γ(R[M]))≦3.We describe all the possibilities for the pair diam(Γ(R))and diam(Γ(R[M])),strictly in terms of the properties of a ring R,where K is a reversible ring and M is a unique product monoid.Moreover,an example showing the necessity of our assumptions is provided.