In this book, we introduce several graphical representations of a ring. In the past three
decades, graphs constructed from algebraic structures have been studied extensively by …

In this paper, we study the strong metric dimension of zero-divisor graph ${\Gamma}(R) $
associated to a ring R. This is done by transforming the problem into a more well-known …

Let R be a commutative ring with identity and M an R-module. In this paper, we associate a
graph to M, say Γ (M), such that when M= R, Γ (M) is exactly the classic zero-divisor graph …

Let R be a commutative ring with unity 1≠ 0. The annihilating-ideal graph of R, denoted by
𝔸 𝔾 (R), is defined to be the graph with vertex set A∗(R)—the set of non-zero annihilating …

Let M be an R-module, where R is a commutative ring with identity 1 and let G (V, E) be a
graph. In this paper, we study the graphs associated with modules over commutative rings …

The annihilating-ideal graph of a commutative ring R is denoted by AG (R), whose vertices
are all nonzero ideals of R with nonzero annihilators and two distinct vertices I and J are …

In this paper, we will extend the notion of zero-divisor graph of commutative rings to zero-
divisor graph of abelian groups and study this zero-divisor graph. We characterize the zero …

Let $ R $ be a ring and $ M $ be an $ R $-module. Two modules $ A $ and $ B $ are called
orthogonal, written $ A\perp B $, if they do not have non-zero isomorphic submodules. We …

We show that an $ R $-module $ M $ is noetherian (resp., artinian) if and only if its
annihilating submodule graph, $\mathbb {G}(M) $, is a non-empty graph and it has …

Let R be a commutative ring with identity and M an R-module. In this paper, we associate a
graph to M, sayΓ (RM), such that when M= R, Γ (RM) coincide with the zero-divisor graph of …