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 $\alpha=a_{1}g_{1}+\cdots‎+ ‎a_{n}g_{n}$ and $\beta=b_{1}h_{1}+\cdots+b_{m}h_{m}$ are adjacent if and only if $a_{i}b_{j}=0$ or $b_{j}a_{i}=0$ 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 $\Gamma(R[M])$‎, ‎where $R$ is a reversible ring and $M$ is a unique product monoid‎.