Vertices of given degree in a random graph

This paper concerns the degree sequence d 1 ≥ d 2 ≥ … ≥ d n of a randomly labeled graph of order n in which the probability of an edge is p(n) ≦ 1/2. Among other results the following questions are answered. What are the values of p(n ) for which d 1 , the maximum degree, is the same for almost every graph? For what values of p(n ) is it true that d 2 > d 2 for almost every graph, that is, there is a unique vertex of maximum degree? The answers are (essentially) p(n ) = o( logn/n/n ) and p(n)n/logn → ∞. Also included is a detailed study of the distribution of degrees when 0 < lim n p(n)/log n ≦ lim n p(n)/log n < ∞.