Mudaber, Mohammad Hassan and Sarmin, Nor Haniza and Gambo, Ibrahim (2022) Perfect codes in the spanning and induced subgraphs of the unity product graph. Mathematics and Statistics, 10 (4). pp. 754758. ISSN 23322071

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Official URL: http://dx.doi.org/10.13189/ms.2022.100406
Abstract
The unity product graph of a ring R is a graph which is obtained by setting the set of unit elements of R as the vertex set. The two distinct vertices ri and rj are joined by an edge if and only if ri · rj = e. The subgraphs of a unity product graph which are obtained by the vertex and edge deletions are said to be its induced and spanning subgraphs, respectively. A subset C of the vertex set of induced (spanning) subgraph of a unity product graph is called perfect code if the closed neighbourhood of c, S1 (c) forms a partition of the vertex set as c runs through C. In this paper, we determine the perfect codes in the induced and spanning subgraphs of the unity product graphs associated with some commutative rings R with identity. As a result, we characterize the rings R in such a way that the spanning subgraphs admit a perfect code of order cardinality of the vertex set. In addition, we establish some sharp lower and upper bounds for the order of C to be a perfect code admitted by the induced and spanning subgraphs of the unity product graphs.
Item Type:  Article 

Uncontrolled Keywords:  commutative ring, induced subgraph, perfect code 
Subjects:  Q Science > QA Mathematics 
Divisions:  Science 
ID Code:  98752 
Deposited By:  Narimah Nawil 
Deposited On:  02 Feb 2023 08:21 
Last Modified:  02 Feb 2023 08:21 
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