Zero-divisor graph

In mathematics, and more specifically in combinatorial commutative algebra, a zero-divisor graph is an undirected graph representing the zero divisors of a commutative ring. It has elements of the ring as its vertices, and pairs of elements whose product is zero as its edges.
Definition.
There are two variations of the zero-divisor graph commonly used.
In the original definition of , the vertices represent all elements of the ring. In a later variant studied by , the vertices represent only the zero divisors of the given ring.
Examples.
If formula_1 is a semiprime number (the product of two prime numbers)
then the zero-divisor graph of the ring of integers modulo formula_1 (with only the zero divisors as its vertices) is either a complete graph or a complete bipartite graph.
It is a complete graph formula_3 in the case that formula_4 for some prime number formula_5. For in this case the vertices are all the nonzero multiples of formula_5, and the product of any two of these numbers is 0 modulo formula_7.
It is a complete bipartite graph formula_8 in the case that formula_9 for two distinct prime numbers formula_5 and formula_11. The two sides of the bipartition are the formula_12 nonzero multiples of formula_11 and the formula_14 nonzero multiples of formula_5, respectively. Two numbers (that are not themselves zero modulo formula_1) multiply to zero modulo formula_1 if and only if one is a multiple of formula_5 and the other is a multiple of formula_11, so this graph has an edge between each pair of vertices on opposite sides of the bipartition, and no other edges. More generally, the zero-divisor graph is a complete bipartite graph for any ring that is a product of two integral domains.
The only cycle graphs that can be realized as zero-product graphs (with zero divisors as vertices) are the cycles of length 3 or 4.
The only trees that may be realized as zero-divisor graphs are the stars (complete bipartite graphs that are trees) and the five-vertex tree formed as the zero-divisor graph of formula_20.
Properties.
In the version of the graph that includes all elements, 0 is a universal vertex, and the zero divisors can be identified as the vertices that have a neighbor other than 0.
Because it has a universal vertex, the graph of all ring elements is always connected and has diameter at most two. The graph of all zero divisors is non-empty for every ring that is not an integral domain. It remains connected, has diameter at most three, and (if it contains a cycle) has girth at most four.
The zero-divisor graph of a ring that is not an integral domain is finite if and only if the ring is finite. More concretely, if the graph has maximum degree formula_21, the ring has at most formula_22 elements.
If the ring and the graph are infinite, every edge has an endpoint with infinitely many neighbors.
 conjectured that (like the perfect graphs) zero-divisor graphs always have equal clique number and chromatic number. However, this is not true; a counterexample was discovered by .