On cycles in the sequence of unitary Cayley graphs

For n is an element of N let p(k)(n) be the number of induced k-cycles in the Cayley graph Cay (Z(n), U-n)(n) where Z(n) is the ring of integers mod n and U-n = Z(n)(*) is the group of units mod n. Our main result is: Given r is an element of N there is a number m(r), depending only on r, with r ln r less than or equal to m(r) less than or equal to 9r! such that P-k(n) = 0 if k greater than or equal to m(r) and n has at most r prime divisors. As a corollary we deduce the existence of non-trivial arithmetic functions f with the properties: f is a Z-finear combination of multiplicative arithmetic functions. f (n) = 0 for every n with at most r different prime divisors. We also prove the chromatic uniqueness of Cay (Z(n), U-n) for n a prime power. (C) 2004 Published by Elsevier B.V.