Non-Trivial Subring Perfect Codes in Unit Graph of Boolean Rings

Mudaber, Mohammad Hassan; Sarmin, Nor Haniza; Gambo, Ibrahim

doi:10.11113/mjfas.v18n3.2503

Cited by 2 publications

(1 citation statement)

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“…A Boolean ring is a ring with identity in which every element is idempotent. Perfect codes in the unit graph of Boolean rings were investigated in [257,258], where the existence of a subring perfect code in the unit graphs associated with the finite Boolean rings was proven in [257], along with a necessary and sufficient condition for a subring of an infinite Boolean ring to admit a perfect code of size infinity in the unit graph. In [258], the perfect codes spanning subgraphs of a unit graph associated with a Boolean ring R of order 2 k , for some positive integer k ≥ 1, were determined, and, as a consequence of this, sharp lower and upper bounds for the cardinality of a subset of the vertex set to be a perfect code spanning subgraphs of a unit graph were established.…”

Section: Unit Graph Of a Ringmentioning

confidence: 99%

Graphs Defined on Rings: A Review

Madhumitha,

Naduvath

2023

Mathematics

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The study on graphs emerging from different algebraic structures such as groups, rings, fields, vector spaces, etc. is a prominent area of research in mathematics, as algebra and graph theory are two mathematical fields that focus on creating and analysing structures. There are numerous studies linking algebraic structures and graphs, which began with the introduction of Cayley graphs of groups. Several algebraic graphs have been defined on rings, a fast-growing area in the literature. In this article, we systematically review the literature on some variants of Cayley graphs that are defined on rings and highlight the properties and characteristics of such graphs, to showcase the research in this area.

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