On Simple Graphs Arising from Exponential Congruences

Abstract EN

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

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 Γϕ(n) of the simple graph G(n) is complete.