Let D(n,r) be a random r-out regular directed multigraph on the set of vertices 1,...,n . In this work, we establish that for every r � 2, there existsr > 0 such that diam(D(n,r)) = (1 + �r + o(1))log r n. Our techniques also allow us to bound some extremal quantities related to the stationary distribution of a simple random walk on D(n,r). In particular, we determine the asymptotic behaviour ofmax andmin, the maximum and the minimum values of the stationary distribution. We show that with
We study properties of a simple random walk on the random digraph D"n","p when np=dlogn, d>1. We prove that whp the value @p"v of the stationary distribution at vertex v is asymptotic to deg^-(v) m where deg^-(v) is the in-degree of v and m=n(n-1)p is the expected number of edges of D"n","p. If d=d(n)-> with n, the stationary distribution is asymptotically uniform whp. Using this result we prove that, for d>1, whp the cover time of D"n","p is asymptotic to dlog(d (d-1))nlogn. If d=d(n)-> with n, then the cover time is asymptotic to nlogn.