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 ax ≡ b (mod n) is solvable. let n = p1k1p2 k2 ⋯ pr kr 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=1 r (φ(pi ki) + 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.