Perfect Codes Over Induced Subgraphs of Unit Graphs of Ring of Integers Modulo n

Abstract

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

Graphs Defined on Rings: A Review

“…It was shown, in [220], that the subgraph of X + n induced by the idempotent elements of the ring Z n admits a perfect code of size two if n is a product of two prime powers, where one of the primes is even; a perfect code of size one if n is the product of k factors of odd prime powers; and a perfect code of size 2 t−1 for the unitary addition Cayley graph on a ring which is the direct product of the factors of Z p k .…”

“…For a graph G, S ⊆ V(G) is a perfect code (different from the notions of a code of a graph) of the graph if S is an independent set, such that every vertex in V(G) − S is adjacent to exactly one vertex in S (see [219]). The perfect codes in an induced subgraph of the unitary addition Cayley graph containing the vertices that represent the idempotent elements of the ring Z n were examined in [220], where the question of when a subset of the idempotent elements in the ring Z n are a perfect code in this induced subgraph of a unitary addition Cayley graph was answered.…”

Perfect Codes in the Spanning and Induced Subgraphs of the Unity Product Graph

“…In recent years, the research of perfect codes in algebraic graphs got much attention and this notion have been extensively investigated in Cayley graphs of groups [16,17,18,19], power and power reduced power graphs of groups [20,21]. However, there are limited number of researches concentrating on investigating the perfect codes in graphs of rings (see, [22,23]).…”