On Simple Graphs Arising from Exponential Congruences

M. Aslam Malik, M. Khalid Mahmood
2012 Journal of Applied Mathematics  
We introduce and investigate a new class of graphs arrived from exponential congruences. For each pair of positive integersaandb, letG(n)denote the graph for whichV={0,1,...,n−1}is the set of vertices and there is an edge betweenaandbif the congruenceax≡b (mod n)is solvable. Letn=p1k1p2k2⋯prkrbe the prime power factorization of an integern, wherep1<p2<⋯<prare distinct primes. The number of nontrivial self-loops of the graphG(n)has been determined and shown to be equal to∏i=1r(ϕ(piki)+1). It is
more » ... (ϕ(piki)+1). It is shown that the graphG(n)has2rcomponents. Further, it is proved that the componentΓpof the simple graphG(p2)is a tree with root at zero, and ifnis a Fermat's prime, then the componentΓϕ(n)of the simple graphG(n)is complete.
doi:10.1155/2012/292895 fatcat:mdnjt2zm7vak5dbokuxwwzft7a