ON THE DIAMETER AND GIRTH OF A ZERO-DIVISOR GRAPH

By David F. Anderson, Shashikant Mulay. 2007

187 citations recorded by the source corpus.

Abstract For a commutative ring R with zero-divisors Z ( R ) , the zero-divisor graph of R is Γ ( R ) = Z ( R ) − { 0 } , with distinct vertices x and y adjacent if and only if x y = 0 . In this paper, we characterize when either diam ( Γ ( R ) ) ≤ 2 or gr ( Γ ( R ) ) ≥ 4 . We then use these results to investigate the diameter and girth for the zero-divisor graphs of polynomial rings, power series rings, and idealizations.

Subfields: Algebra and Number Theory, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/e114fe22a6502e6cbde361524c59ca0ef421d156

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Citations in the corpus

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]).”
The total graph of a commutative ring (2008, background): “The zero-divisor graph Γ (R(+)M) has recently been studied in [4] and [6].”
Commutative Boolean monoids, reduced rings, and the compressed zero-divisor graph (2012, background): “2, together with characterizations of gr(Γ (R)) from [6], we can easily show that actually gr(ΓE(R)) ∈ {3,∞}.”
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).”
THE GENERALIZED TOTAL GRAPH OF A COMMUTATIVE RING (2013, background): “The zero-divisor graph Γ(R(+)M) has been studied in [10, 13], and the total graph T (Γ(R(+)M)) has been studied in [23].”
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).”