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 n = p1k1p2k2 ⋯ prkr 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 ∏i=1r (φ(piki) + 1). 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 Γ φ(n) of the simple graph G(n) is complete. © 2012 M. Aslam Malik and M. Khalid Mahmood.