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.

The induced subgraph of a unit graph with vertex set as the idempotent elements of a ring R is a graph which is obtained by deleting all non idempotent elements of R. Let C be a subset of the vertex set in a graph Γ. Then C is called a perfect code if for any x, y ∈ C the union of the closed neighbourhoods of x and y gives the the vertex set and the intersection of the closed neighbourhoods of x and y gives the empty set. In this paper, the perfect codes in induced subgraphs of the unit graphs associated with the ring of integer modulo n, Zn that has the vertex set as idempotent elements of Zn are determined. The rings of integer modulo n are classified according to their induced subgraphs of the unit graphs that accept a subset of a ring Zn of different sizes as the perfect codes

No abstract

The induced subgraph of a unit graph with vertex set as the non unit elements of a ring R is a graph obtained by deleting all unit elements of R. In a graph 􀀀, a subset of the vertex set is called a perfect code if the balls with radius 1 centred on the subset are pairwise disjoint and their unions yield the whole vertex set. In this paper, we determine the perfect codes of induced subgraphs of the unit graphs associated with some finite commutative rings R with unity that has a vertex set as non unit elements of R. Moreover, we classify the commutative rings in which their associated induced subgraphs of unit graphs admit the trivial and non-trivial perfect codes. We also characterize the commutative rings based on the induced subgraph of unit graphs that do not admit the perfect codes. Furthermore, we prove that the complement induced subgraph of unit graph admit only the trivial subring perfect code.

The aim of this paper is to investigate the non-trivial subring perfect codes in a unit graph associated with the Boolean rings. We prove a subring perfect code of size , where , in the unit graphs associated with the finite Boolean rings . Moreover, we give a necessary and sufficient condition for a subring of an infinite Boolean ring  to be a perfect code of size infinity in the unit graph.