The degree sequence of a random graph. I. The models
Random Structures & Algorithms
Asymptotic Enumeration of Graphs with a Given Upper Bound on the Maximum Degree
Combinatorics, Probability and Computing
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Let 0 p q = 1 − p and b be fixed and let G ∈ 𝒢(n, p) be a graph on n vertices where each pair of vertices is joined independently with probability p. We show that the probability that every vertex of G has degree at most pn + b √npq is equal to (c + o(1))n, for c = c(b) the root of a certain equation. Surprisingly, c(0) = 0.6102 … is greater than ½ and c(b) is independent of p. To obtain these results we consider the complete graph on n vertices with weights on the edges. Taking these weights as independent normal N(p, pq) random variables gives a ‘continuous’ approximation to 𝒢(n, p) whose degrees are much easier to analyse.