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 …

Let S be a (multiplicative) commutative semigroup with 0. Associate to S a (simple) graph G
(S) with vertices the nonzero zero-divisors of S, and two distinct vertices x and y are adjacent …

Suppose that R is a commutative ring with identity. Let A (R) be the set of all ideals of R with
non-zero annihilators. The annihilating-ideal graph of R is defined as the graph AG (R) with …

Let A be a commutative ring with nonzero identity, 1≤ n<∞ be an integer, and R= A× A×…×
A (n times). The total dot product graph of R is the (undirected) graph TD (R) with vertices …

Let R be a commutative ring and 𝔸 (R) be the set of ideals with non-zero annihilators. The
annihilating-ideal graph of R is defined as the graph 𝔸𝔾 (R) with the vertex set 𝔸 (R)*= 𝔸 …

Let R be a commutative ring and Z (R) be its set of all zero-divisors. The total graph of R,
denoted by T Γ (R), is the undirected graph with vertex set R and two distinct vertices x and y …

In this paper, a new kind of graph on a commutative ring R with identity, namely the co-
maximal ideal graph is defined and studied. We use to denote this graph, with its vertices the …

Let R be a ring with unity. The upper ideal relation graph Γ U (R) of the ring R is the simple
undirected graph whose vertex set is the set of all non-unit elements of R and two distinct …

Let R be a commutative ring with identity which is not an integral domain. An ideal I of a ring
R is called an annihilating ideal if there exists r∈ R\{0} such that Ir=(0). In this paper, we …

Let R be a commutative ring with identity and A∗(R) the set of nonzero ideals with non-zero
annihilators. The annihilating-ideal graph of R is defined as the graph AG (R) with the vertex …