Graph-Theoretic Insights into Digraph Complement Γ̅(n, 2)

Abstract

For each positive integer n, we assign a power digraph modulo n, denoted by Γ(n, 2), whose vertex set is ℤₙ = {0, 1, 2, ..., n - 1}, and for which there is a directed edge from a vertex a to a vertex b if and only if a² ≡ b (mod n), where a, b ∈ ℤₙ. We investigate the graph-theoretic properties of the complement of the digraph Γ(n, 2), denoted Γ̅(n, 2). In particular, we study fixed points, isolated points, in-degree, and out-degree of the vertices of the digraph Γ̅(n, 2).