A Kind of Graph Structure Associated With Zero-Divisors of Monoid Rings

Let R be an associative ring and M be a monoid‎. ‎In this paper‎, ‎we introduce new kind of graph structure asociated with zero-divisors of monoid ring $R[M]$‎, ‎calling it the M-Armendariz graph of a ring R and denoted by $A(R,M)$‎. ‎It is an undirected graph whose vertices are all non-zero zero-divisors of the monoid ring $R[M]$ and two distinct vertices α=a₁g₁+⋯‎+ ‎aₙgₙ and β=b₁h₁+⋯+bₘhₘ are adjacent if and only if aᵢbⱼ=0 or bⱼaᵢ=0 for all $i,j$‎. ‎We investigate some graph properties of $A(R,M)$ such as diameter‎, ‎girth‎, ‎domination number and planarity‎. ‎Also‎, ‎we get some relations between diameters of the M-Armendariz graph $A(R,M)$ and that of zero divisor graph Γ(R[M])‎, ‎where R is a reversible ring and M is a unique product monoid‎.