A Structural Investigation of the Idempotent Graph Associated with the Ring Zpq

Abstract

This study investigates the structural characteristics of the idempotent graph G_Id (R) associated with a commutative ring R. The graph is defined as a simple, undirected graph in which the vertices correspond to the elements of R, and two distinct vertices a and b are considered adjacent precisely when the condition 〖 (a+b)〗^2=a+b holds, meaning their sum is an idempotent element. The research specifically examines the idempotent graph constructed over the ring of integers modulo n, denoted 〖Z〗_n, where n=pq and p and q are distinct prime numbers satisfying p<q. Within this framework, the work provides a detailed analysis of several fundamental graph-theoretic properties of G_Id (R), including its radius, diameter, vertex degree, and chromatic number. The findings offer insight into how the algebraic structure of 〖 Z〗_pq influences the resulting graph, contributing to a deeper understanding of the interaction between ring theory and graph theory.