On Simple Graphs Arising from Exponential Congruences

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 a ≡ b modn is solvable. Let n p1 1 p k2 2 · · · pr r 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 ∏r i 1 φ p ki i 1 . It is shown that the graph G n has 2 r 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.