1. Induced subgraphs of zero-divisor graphs
- Author
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Arunkumar, G., Cameron, Peter J., Kavaskar, T., Chelvam, T. Tamizh, University of St Andrews. Pure Mathematics, and University of St Andrews. Centre for Interdisciplinary Research in Computational Algebra
- Subjects
Q Science ,Mathematics(all) ,Universal graph ,Rado graph ,T-NDAS ,Local ring ,Mathematics - Rings and Algebras ,16B99 ,Rings and Algebras (math.RA) ,MCP ,FOS: Mathematics ,Mathematics - Combinatorics ,Combinatorics (math.CO) ,QA Mathematics ,Zero divisor ,QA - Abstract
Funding: Peter J. Cameron acknowledges the Isaac Newton Institute for Mathematical Sciences, Cambridge, for support and hospitality during the programme Groups, representations and applications: new perspectives (supported by EPSRC grant no. EP/R014604/1), where he held a Simons Fellowship. For this research, T. Kavaskar was supported by the University Grant Commissions Start-Up Grant, Government of India grant No. F. 30-464/2019 (BSR) dated 27.03. T. Tamizh Chelvam was supported by CSIR Emeritus Scientist Scheme (No. 21 (1123)/20/EMR-II) of Council of Scientific and Industrial Research, Government of India. 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 ab=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. Publisher PDF
- Published
- 2023