Applications of Solutions of Difference Equations in Graph Theory

Abstract

Graph labeling, which assigns values to the vertices and edges of a graph under specific conditions, has significant applications in real-world problems such as coding theory, radar code design, synch-set codes, missile guidance, and convolution codes with optimal error-correction properties. This study explores the connections between graph labeling and solutions of difference equations by constructing infinite graphs from sequences of real or complex numbers. Each solution of a difference equation induces a labeled graph in the complex plane, where vertex functions extend naturally to edge functions through binary operations over the complex field. Furthermore, the use of complex plane labeling provides a framework for visualizing higher-dimensional relationships in two-dimensional settings, enriching the structural understanding of labeled graphs and their diverse applications.