The diameter of a zero divisor graph

By T. Lucas. 2006

177 citations recorded by the source corpus.

Let R be a commutative ring and let Z(R)∗ be its set of nonzero zero divisors. The set Z(R)∗ makes up the vertices of the corresponding zero divisor graph, Γ(R), with two distinct vertices forming an edge if the product of the two elements is zero. The distance between vertices a and b (not necessarily distinct from a) is the length of the shortest path connecting them, and the diameter of the graph, diam(Γ(R)), is the sup of these distances. For a reduced ring R with nonzero zero divisors, 1⩽diam(Γ(R))⩽diam(Γ(R[x]))⩽diam(Γ(R〚x〛))⩽3. A complete characterization for the possible diameters is given exclusively in terms of the ideals of R. A similar characterization is given for diam(Γ(R)) and diam(Γ(R[x])) when R is nonreduced. Various examples are provided to illustrate the difficulty in dealing with the power series ring over a nonreduced ring.

Subfields: Algebra and Number Theory, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/b82b4376e53af15fee6a44467857ffa57ba10622

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Citations in the corpus

The total graph of a commutative ring (2008, background): “For some other recent papers on zero-divisor graphs, see [1,2,4–6,12–14].”
A Zero Divisor Graph Determined by Equivalence Classes of Zero Divisors (2007, background): “Example 3.7. Recall that the graph EG(Z12) in Corollary 1.11 had two vertices, namely [ 6 ] and [4], of maximal degree.”
On the Dot Product Graph of a Commutative Ring (2015, unknown): “Recently, there has been considerable attention in the literature to associating graphs with algebraic structures (see [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [17], [18], [19], [20], [21], [23], [24], [25], and [26]).”
ON THE DIAMETER AND GIRTH OF A ZERO-DIVISOR GRAPH (2007, methodology): “Some cases for R a non-noetherian commutative ring left open in [4] were then resolved in [12].”
The Annihilating-Ideal Graph of Commutative Rings I (2008, background): “The zero divisor graph of R, denoted by Γ(R), is a graph with the vertex set Z(R)∗ := Z(R) in which for every two vertices x and y, x−−−y is an edge if and only if x 6= y and xy = 0.”
Zero-divisor graphs of amalgamated duplication of a ring along an ideal (2007, background): “More recently, Axtell, Coykendall and Stickles [5] and Luca s [10] have studied the diameter of the corresponding graphs of the polynomial ring R[x] and the power series ringR[[x]].”
Zero-Divisor Graphs of Matrices Over Commutative Rings (2009, background): “Recently, the diameter and girth of polynomial and power series rings over a commutative ring were studied in Axtell et al. (2005), Anderson and Mulay (2007), and Lucas (2006).”
The Annihilating-Ideal Graph of Commutative Rings II (2008, methodology): “Moreover, Lucas in [6] characterized the diameter of Γ(R) in terms of the ideals of R.”
On the Zero-Divisor Graph of a Ring (2008, background): “For other articles on zero-divisor graphs, see Anderson and Naseer (1993), Anderson (2008), Anderson et al. (2001, 2003), Anderson and Mulay (2007), Axtel et al. (2005), Axtel and Stickles (2006), DeMeyer and Schneider (2002), Lucas (2006), Mulay (2002), Redmond (2007), and Smith (2003).”
On the Annihilator Graph of a Commutative Ring (2014, background): “Since commutative rings with exactly one nonzero zero-divisor are studied in [2, 3, 10], throughout this article we only consider commutative rings with at least two nonzero zero-divisors.”