Mathematics and Statistics Vol. 10(4), pp. 754 - 758
DOI: 10.13189/ms.2022.100406
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Perfect Codes in the Spanning and Induced Subgraphs of the Unity Product Graph

Mohammad Hassan Mudaber ^*, Nor Haniza Sarmin , Ibrahim Gambo
Department of Mathematical Sciences, Faculty of Science, Universiti Teknologi Malaysia, 81310 UTM Johor Bahru, Malaysia

ABSTRACT

The unity product graph of a ring is a graph which is obtained by setting the set of unit elements of as the vertex set. The two distinct vertices and are joined by an edge if and only if . 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 of the vertex set of induced (spanning) subgraph of a unity product graph is called perfect code if the closed neighbourhood of , forms a partition of the vertex set as runs through . In this paper, we determine the perfect codes in the induced and spanning subgraphs of the unity product graphs associated with some commutative rings with identity. As a result, we characterize the rings 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 to be a perfect code admitted by the induced and spanning subgraphs of the unity product graphs.

KEYWORDS
Commutative Ring, Unity Product Graph, Induced Subgraph, Spanning Subgraph, Perfect Code

Cite This Paper in IEEE or APA Citation Styles
(a). IEEE Format:
[1] Mohammad Hassan Mudaber , Nor Haniza Sarmin , Ibrahim Gambo , "Perfect Codes in the Spanning and Induced Subgraphs of the Unity Product Graph," Mathematics and Statistics, Vol. 10, No. 4, pp. 754 - 758, 2022. DOI: 10.13189/ms.2022.100406.

(b). APA Format:
Mohammad Hassan Mudaber , Nor Haniza Sarmin , Ibrahim Gambo (2022). Perfect Codes in the Spanning and Induced Subgraphs of the Unity Product Graph. Mathematics and Statistics, 10(4), 754 - 758. DOI: 10.13189/ms.2022.100406.