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The main result of this paper is the fact that the fraction of primes p ≤ x satisfying the condition that p − 1 has a prime divisor q > exp(ln x/ln ln x) and the number of prime divisors of q − 1 essentially differ from ln ln( x/n), where n = ( p − 1)/ q, tends to zero as x increases. [ABSTRACT FROM AUTHOR]

Let p be an odd prime. In this paper we prove that all tetravalent connected Cayley graphs of order p3 are normal. As an application, a classification of tetravalent symmetric graphs of odd prime–cube order is given. [ABSTRACT FROM AUTHOR]