Shahrood University of TechnologyJournal of Algebraic Systems2345-512811220240101A KIND OF GRAPH STRUCTURE ASSOCIATED WITH ZERO-DIVISORS OF MONOID RINGS5363273010.22044/jas.2022.12238.1646ENMohammad EtezadiDepartment of Pure Mathematics, Faculty of Mathematical Sciences, University of
Tabriz, Tabriz, Iran.Abdollah AlhevazFaculty of Mathematical Sciences, Shahrood University of Technology, P.O. Box
316-3619995161, Shahrood, Iran0000-0001-6167-607XJournal Article20220828Let $R$ 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.https://jas.shahroodut.ac.ir/article_2730_aa5f7f98a22a1436fed3a41607d47007.pdf