The zero-divisor graph of ${\displaystyle \mathbb {Z} _{2}\times \mathbb {Z} _{4}}$, the only possible zero-divisor graph that is a tree but not a star

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.^[1]

Definition

There are two variations of the zero-divisor graph commonly used. In the original definition of Beck (1988), the vertices represent all elements of the ring.^[2] In a later variant studied by Anderson & Livingston (1999), the vertices represent only the zero divisors of the given ring.^[3]

Examples

If ${\displaystyle n}$ is a semiprime number (the product of two prime numbers) then the zero-divisor graph of the ring of integers modulo ${\displaystyle n}$ (with only the zero divisors as its vertices) is either a complete graph or a complete bipartite graph. It is a complete graph ${\displaystyle K_{p-1}}$ in the case that ${\displaystyle n=p^{2}}$ for some prime number ${\displaystyle p}$. In this case the vertices are all the nonzero multiples of ${\displaystyle p}$, and the product of any two of these numbers is zero modulo ${\displaystyle p^{2}}$.^[3]

It is a complete bipartite graph ${\displaystyle K_{p-1,q-1}}$ in the case that ${\displaystyle n=pq}$ for two distinct prime numbers ${\displaystyle p}$ and ${\displaystyle q}$. The two sides of the bipartition are the ${\displaystyle p-1}$ nonzero multiples of ${\displaystyle q}$ and the ${\displaystyle q-1}$ nonzero multiples of ${\displaystyle p}$, respectively. Two numbers (that are not themselves zero modulo ${\displaystyle n}$) multiply to zero modulo ${\displaystyle n}$ if and only if one is a multiple of ${\displaystyle p}$ and the other is a multiple of ${\displaystyle q}$, 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.^[3]

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.^[3] 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 ${\displaystyle \mathbb {Z} _{2}\times \mathbb {Z} _{4}}$.^[1][3]

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,^[3] and (if it contains a cycle) has girth at most four.^[4][5]

The zero-divisor graph of a ring that is not an integral domain is finite if and only if the ring is finite.^[3] More concretely, if the graph has maximum degree ${\displaystyle d}$, the ring has at most ${\displaystyle (d^{2}-2d+2)^{2}}$ elements. If the ring and the graph are infinite, every edge has an endpoint with infinitely many neighbors.^[1]

Beck (1988) 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 Anderson & Naseer (1993).^[6]

References