Ideal based trace graph of matrices

Chelvam, T. Tamizh; Manikandan, Sivagami

doi:10.15672/hujms.478373

Cited by 6 publications

(1 citation statement)

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“…Another very general problem is to examine the induced subgraphs of various generalizations of zero-divisor graphs, such as the extended zerodivisor graphs and the trace graphs [2,24,25,26,27].…”

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confidence: 99%

Induced subgraphs of zero-divisor graphs

G.¹,

Cameron²,

Kavaskar³

et al. 2022

Preprint

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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.

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confidence: 99%