Perfect Codes in Induced Subgraph of Unit Graph Associated With Some Commutative Rings

Abstract

The unit graph associated with a ring is the graph whose vertices are elements of , and two different vertices and are adjacent if and only if where is the set of unit elements of The aim of this paper is to present the perfect codes in induced subgraph of unit graph associated with some commutative rings with unity in which its vertex set is We characterize some families of commutative rings with induced subgraphs of unit graphs accepting the non-trivial perfect codes, and some other families of commutative rings with induced subgraphs of unit graphs which do not accept perfect codes.

Graphs Defined on Rings: A Review

“…Furthermore, it was proven that the complement of this induced subgraph of the unit graph of finite commutative rings admits only the trivial subring perfect code, where a subring perfect code refers to the perfect code on a subgraph induced by a subring of the ring. A similar investigation on some other induced subgraphs of the unit graph of commutative rings was conducted in [256], whose results were analogous to the ones obtained in [255], even though the vertex set of the induced subgraphs differed. This gives an underlying property of the unit graph of the ring itself rather than the subgraphs.…”

“…[219]). By the definition of a perfect code, the investigation of perfect codes can be seen as computing a variant of the domination number of a graph, and, in [254], perfect codes in the unit graphs were examined, where the rings were characterised first based on the existence of a perfect code in their unit graphs or their complements, as finding whether a graph admits perfect code is also a question that remains to be addressed. Following this characterisation of rings, the commutative rings with identities in which their associated unit graphs accepted perfect codes of order one and two were characterised, and a few results relating the structure of the perfect code and the structure of the rings were obtained.…”

“…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.…”