An exploration of ideal-divisor graphs

Abstract

Zero-divisor graphs have given some interesting insights into the behavior of commutative rings. Redmond introduced a generalization of the zero-divisor graph called an ideal-divisor graph. This paper expands on Redmond's findings in an attempt to find additional information about the structure of commutative rings from ideal-divisor graphs. Definitions and introductionThroughout, we assume that R is a finite commutative ring with identity, though in some instances the proofs given can be extended to more general rings. A zerodivisor in R is an element x such that there exists a nonzero y ∈ R with x y = 0. The set of all zero-divisors in R is denoted by Z (R). The set of all nonzero zero-divisors is denoted by Z (R) * .A graph G is defined by a vertex set V (G) and an edge setTwo vertices x and y joined by an edge are said to be adjacent, denoted x − y. A vertex x is said to be looped if x − x. A path between two elements a 1 , a n ∈ V (G) is an ordered sequence {a 1 , a 2 , . . . , a n } of distinct vertices of G such that a i−1 − a i for all 1 < i ≤ n. If there exists a path between any two distinct vertices, then the graph is said to be connected. A graph is said to be complete if every vertex is adjacent to every other vertex, and we denote the complete graph on n vertices byIf the vertices of a graph G can be partitioned into two sets with vertices adjacent only if they are in distinct sets, then G is bipartite. If vertices in a bipartite graph are adjacent if and only if they are in distinct vertex sets, then the graph is called complete bipartite. We will denote the complete bipartite graph with distinct vertex sets of cardinalities m and n by K m,n . A star graph is a complete bipartite graph MSC2010: 13M05.