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 Beck (1988), the vertices represent all elements of the ring. In a later variant studied by Anderson & Livingston (1999), the vertices represent only the zero divisors of the given ring.
Examples If n {\displaystyle n} is a semiprime number (the product of two prime numbers) then the zero-divisor graph of the ring of integers modulo n {\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 K p − 1 {\displaystyle K_{p-1}} in the case that n = p 2 {\displaystyle n=p^{2}} for some prime number p {\displaystyle p} . In this case the vertices are all the nonzero multiples of p {\displaystyle p} , and the product of any two of these numbers is zero modulo p 2 {\displaystyle p^{2}} . It is a complete bipartite graph K p − 1 , q − 1 {\displaystyle K_{p-1,q-1}} in the case that n = p q {\displaystyle n=pq} for two distinct prime numbers p {\displaystyle p} and q {\displaystyle q} . The two sides of the bipartition are the p − 1 {\displaystyle p-1} nonzero multiples of q {\displaystyle q} and the q − 1 {\displaystyle q-1} nonzero multiples of p {\displaystyle p} , respectively. Two numbers (that are not themselves zero modulo n {\displaystyle n} ) multiply to zero modulo n {\displaystyle n} if and only if one is a multiple of p {\displaystyle p} and the other is a multiple of q {\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. 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 Z 2 × Z 4 {\displaystyle \mathbb {Z} _{2}\times \mathbb {Z} _{4}} .
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 d {\displaystyle d} , the ring has at most ( d 2 − 2 d + 2 ) 2 {\displaystyle (d^{2}-2d+2)^{2}} elements. If the ring and the graph are infinite, every edge has an endpoint with infinitely many neighbors. 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).
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