On Symmetry of Complete Graphs over Quadratic and Cubic Residues

In this study, we investigate two graphs, one of which has units of a ring Z n as vertices (or nodes) and an edge will be built between two vertices u and v if and only if u 3 ≡ v 3 mod   n . This graph will be termed as cubic residue graph. While the other is called Gaussian quadratic residue graph whose vertices are the elements of a Gaussian ring Z n i of the form α = a + i b , β = c + i    d , where a , b , c , d are the units of Z n . Two vertices α and β are adjacent to each other if and only if α 2 ≡ β 2 mod   n . In this piece of work, we characterize cubic and Gaussian quadratic residue graphs for each positive integer n in terms of complete graphs.