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.
Highlights
Graph theory plays a dynamic role in computer science, biological sciences, chemistry, and physics [1,2,3,4,5,6,7]
Many circuits are constructed in physics with the use of graphs [4]
We elaborate the concept of cubic residue graph and characterize these graphs completely for each positive integer n. e disjoint union of the graphs H and K is expressed by H⊕K, and the disjoint union of the n copies of the graph K is denoted as nK K⊕K⊕ · · · ⊕K
Summary
Graph theory plays a dynamic role in computer science, biological sciences, chemistry, and physics [1,2,3,4,5,6,7]. E concept of square mapping x2 ⟶ y under modulo prime number is discussed by Rogers in [8]. Before introducing results for cubic and Gaussian quadratic residues graphs, we give some earlier results without proofs for use in the sequel. If p is a prime number of the form p ≡ 1(mod 3), the number of different cubic residues in mod p is (p + 2)/3. (2) If p|f′(u) and pk+1|f(u), there are p solutions of f(x) ≡ 0(mod pk+1) that are congruent to u modulo p, given by u + pkv for v 0, 1, 2, . (3) If p|f′(u) and pk+1∤f(u), there are no solutions of f(x) ≡ 0(mod pk+1) that are congruent to u modulo pk Theorem 4 (See [19]). − 1, if a is quadratic residue (mod p), p|a, if a is quadratic nonresidue (mod p). (1)
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