On Symmetry of Complete Graphs over Quadratic and Cubic Residues

Abstract

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.

Exploring a Graph Complement in Quadratic Congruence

“…Castillo and Mainguez, in their 2022 study on modular arithmetic, delve into the symmetrical patterns within sets of k-units modulo n [1]. Mateen et al, in their 2021 research in graph theory, investigate the symmetrical intricacies within complete graphs involving quadratic and cubic residues [2]. Ali, Bahrami, and Reza's 2017 contribution uncovers symmetrical properties in Cayley graphs for the ring of Gaussian integers modulo n [3].…”

Properties of Graph Based on Divisor-Euler Functions

“…Results regarding algorithm and graph labeling were discussed in [9][10][11]. Many results on digraphs and their labeling based on number theory are discussed in [12][13][14][15]. Results on upper bound sequence of networks and RN's via Lambert type Map can be seen in [16][17][18].…”

On Fixed Points of Digraphs Over Lambert Type Map

“…Before finding the fixed point, one should understand the meanings of fixed point with respect to the environment, we are using for. For a detail understanding of fixed points and integer classes, we suggest to read the references [1][2][3], [4,5,17], [18][19][20] and then [6][7][8][9][10][11][12][13][14][15][16], so that the reader could enjoy the reading of our proposed results.A graph is an ordered pair G(V, E) which consists on two sets V and E, where V is set of points named as set of vertices taking as the residues of any given fixed integer and E is a set of edges, obtained by using DLTM and a graph in which edges have direction is called directed graphs or digraphs Definition 1. Graph A graph is an ordered pair G(V, E) which consists on two sets V and E, where V is set of points named as set of vertices and E is a set of edges.…”