On the Asymptotic Behavior of Degrees of Vertices in a Random Graph

Rulers and slaves in a random social group

“…It should be remarked here that in a case when o~(n) tends to infinity very slowly the bounds for d,*~ and d(,_i+l) are complicated since they are expressed by some power series (see [5] and [7] [] Now, let us turn our attention to a case when the out-degree of each point in D(n,d) is of the order of magnitude log n. From now on we shall restrict our considerations to the minimum and maximum in-degree of D(n, d) only. First, note that Erdrs and Rrnyi [4] and Ivchenko [5] proved (using the approach described with a constant ce then with a positive probability there are more than one SLAVE in a social group. Also it is clear that if e¢ = ct(n) is a sequence tending to infinity (arbitrarily slowly) then Prob(d(]) > 0) ~ 1 as n ~ or.…”

The asymptotic value of the Randić index for trees

“…Recall that for almost all graphs the degree of a vertex is (p + o(1))n (see [8]). Thus, for almost all graphs, R(G n,p ) = 1 2 · 1 (p + o(1)) 2 n 2 · (p + o(1))n · n = ( 1 2 + o(1))n.…”

In-degree sequence in a general model of a random digraph

“…Similarly, let F (t; n, p) denote probability of at least t successes in such distribution. In the proofs of our main results we will need a very precise etimate of the asymptotic behaviour of the distribution function of the binomial law with parameters n and p, where p = p(n) = o(1) and np/ log n → ∞ as n → ∞ (see [5] and [12]).…”

“…It is known (see e.g. [5]) that this equation has a negative solution z(u, a) and a positive solution y(u, a), which in some neighbourhood of zero are given by the power series…”