Zero-divisor graphs in commutative rings

The ideal-based zero-divisor graph of commutative chained rings

“…If x, y ∈ N I (R), then xy ∈ I by Proposition 3.1 (2), and thus d(x, y) = 1. If x ∈ N I (R) and y / ∈ N I (R), then yz ∈ I for some z ∈ N I (R) * ⊆ Z I (R) * by Proposition 3.1 (4) and xz ∈ I by Proposition 3.1 (2). If x = z, then d(x, y) = 1.…”

“…Let Z(R) be the set of zero-divisors of R. In [5], D. F. Anderson and P. S. Livingston associated a (simple) graph Γ(R) to R, with vertices Z(R) * = Z(R)\{0}, the set of nonzero zero-divisors of R, and distinct vertices x and y are adjacent if and only if xy = 0. The zero-divisor graph Γ(R) of R has been studied extensively; see the the survey articles [2] and [9].…”

Nonzero Component Graph of a Finite Dimensional Vector Space

“…The benefit of studying these graphs is that one may find some results about the algebraic structures and vice versa. There are three major problems in this area: (1) characterization of the resulting graphs, (2) characterization of the algebraic structures with isomorphic graphs, and (3) realization of the connections between the structures and the corresponding graphs.…”

“…Though his key goal was to address the issue of colouring, this initiated the formal study of exposing the relationship between algebra and graph theory and at advancing applications of one to the other. Till then, a lot of research, e.g., [11,2,3,1,8,6,7,4] has been done in connecting graph structures to various algebraic objects. Recently, intersection graphs associated with subspaces of vector spaces were studied in [10,12].…”

A Generalization of the Zero-Divisor Graph for Modules

“…In recent decades, the zero-divisor graphs of commutative rings (in this paper, called the classic zero-divisor graph) have been extensively studied by many authors and have become a major field of research, see for example [3][4][5][6][7][8][9][10][11][12][13][14][15]. Some authors have also extended the graph of zero-divisors to non-commutative rings, see [18] and [2].…”