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Induced subgraphs of zero-divisor graphs.
- Source :
-
Discrete Mathematics . Oct2023, Vol. 346 Issue 10, pN.PAG-N.PAG. 1p. - Publication Year :
- 2023
-
Abstract
- The zero-divisor graph of a finite commutative ring with unity is the graph whose vertex set is the set of zero-divisors in the ring, with a and b adjacent if a b = 0. We show that the class of zero-divisor graphs is universal, in the sense that every finite graph is isomorphic to an induced subgraph of a zero-divisor graph. This remains true for various restricted classes of rings, including boolean rings, products of fields, and local rings. But in more restricted classes, the zero-divisor graphs do not form a universal family. For example, the zero-divisor graph of a local ring whose maximal ideal is principal is a threshold graph; and every threshold graph is embeddable in the zero-divisor graph of such a ring. More generally, we give necessary and sufficient conditions on a non-local ring for which its zero-divisor graph to be a threshold graph. In addition, we show that there is a countable local ring whose zero-divisor graph embeds the Rado graph , and hence every finite or countable graph, as induced subgraph. Finally, we consider embeddings in related graphs such as the 2-dimensional dot product graph. [ABSTRACT FROM AUTHOR]
Details
- Language :
- English
- ISSN :
- 0012365X
- Volume :
- 346
- Issue :
- 10
- Database :
- Academic Search Index
- Journal :
- Discrete Mathematics
- Publication Type :
- Academic Journal
- Accession number :
- 164962711
- Full Text :
- https://doi.org/10.1016/j.disc.2023.113580