On Simple Graphs Arising from Exponential Congruences

Abstract

We introduce and investigate a new class of graphs arrived from exponential congruences. For each pair of positive integers a and b, let G(n) denote the graph for which V = {0, 1, …, n − 1} is the set of vertices and there is an edge between a and b if the congruence ax ≡ b (mod n) is solvable. Let be the prime power factorization of an integer n, where p1 < p2 < ⋯<pr are distinct primes. The number of nontrivial self‐loops of the graph G(n) has been determined and shown to be equal to . It is shown that the graph G(n) has 2r components. Further, it is proved that the component Γp of the simple graph G(p2) is a tree with root at zero, and if n is a Fermat′s prime, then the component Γϕ(n) of the simple graph G(n) is complete.