On zero-divisor graphs of small finite commutative rings

Abstract

In this article, all graphs on n = 6, 7, . . . , 14 vertices which can be realized as the zero-divisor graphs of a commutative rings with 1, and the list of all rings (up to isomorphism) which produce these graphs, are given. An algorithm is presented to find (up to isomorphism) all commutative, reduced rings with 1 which give rise to a zero-divisor graph on n vertices for any n 1. Also, the zero-divisor graph of a finite commutative ring is used to find bounds on the size of that ring.

Some Properties of the Idempotent Graph of a Ring

“…Many fundamental papers devoted to graphs assigned to a ring have appeared recently, see for example [1][2][3][4][5][6][7][8][9][10][11][12]14,15,17,18].…”

Zero-divisor ideals and realizable zero-divisor graphs

“…A two-vertex graph where neither endpoint is looped can be realized as the graph of ‫ޚ‬ 2 × ‫ޚ‬ 2 . If is a graph on two vertices and only one endpoint is looped, then it is not realizable as a zero-divisor graph, as shown by [Redmond 2007]. …”

“…More information about graph theory may be found in [Wilson 1972]. We define the zero-divisor graph of R, denoted (R), as follows: x ∈ V ( (R)) if and only if x ∈ Z (R) * , and x-y if and only if x y = 0.…”

“…We define the zero-divisor graph of R, denoted (R), as follows: x ∈ V ( (R)) if and only if x ∈ Z (R) * , and x-y if and only if x y = 0. We will allow loops in (R), which is a change from other definitions of zero-divisor graphs as in [Anderson and Livingston 1999;Axtell et al 2006;Lucas 2006;Redmond 2007].…”

On the Annihilator Graph of Group Rings

“…The work of Beck is further continued by Anderson and Naseer in [6] and, for other graph theoretical aspects, by Anderson and Livingston in [5]. While they focus just on the zero-divisors of the rings, there are many other kinds of graphs associated to ring, some of which are extensively studied, see for example [1,2,3,4,5,6,7,9,10,11,14,20].…”