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.

A KIND OF GRAPH STRUCTURE ASSOCIATED WITH ZERO-DIVISORS OF MONOID RINGS Keywords:

‎M-Armendariz graph‎ , ‎diameter‎ , ‎unique product monoid‎ , ‎monoid ring‎ , ‎domination number