Perfect codes in the spanning and induced subgraphs of the unity product graph

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.