Perfect Codes in Graph Theory: A Ring-Theoretic Perspective

Abstract

A graph is a mathematical structure that represents a network described between lines and points. Due to the graph’s many fascinating features, its characteristics are widely studied by many researchers. One of the popular topics studied on graphs is perfect code. A set V of vertices in a graph G is called a perfect code if every vertex in G is either in V or is adjacent to exactly one vertex in V . Originally, perfect codes were used in coding theory. It is then extended to other fields including graph theory. In this paper, mathematical concepts of perfect codes of graphs are bridged using ring theory. The perfect codes are determined for the zero divisor graph of some finite rings of matrices with dimension two. First, the zero divisor graph of the finite rings of matrices is constructed where its vertices are all zero divisors of the ring and two distinct vertices are adjacent if their product is zero. Then, from the graph’s vertex set, the vertices’ neighborhood elements are determined to compute the graph’s perfect codes.