The diameter of a cycle plus a random matching
SIAM Journal on Discrete Mathematics
Random trees and random graphs
proceedings of the eighth international conference on Random structures and algorithms
Regular Article: The Diameter of Sparse Random Graphs
Advances in Applied Mathematics
Proof of the Van den Berg–Kesten Conjecture
Combinatorics, Probability and Computing
The Diameter of a Scale-Free Random Graph
Combinatorica
Counting connected graphs inside-out
Journal of Combinatorial Theory Series B
Distances in random graphs with finite variance degrees
Random Structures & Algorithms
The phase transition in inhomogeneous random graphs
Random Structures & Algorithms
The diameter of sparse random graphs
Random Structures & Algorithms
Diameters in supercritical random graphs via first passage percolation
Combinatorics, Probability and Computing
Anatomy of a young giant component in the random graph
Random Structures & Algorithms
The mixing time of the Newman: Watts small world
Proceedings of the twenty-third annual ACM-SIAM symposium on Discrete Algorithms
Anatomy of the giant component: The strictly supercritical regime
European Journal of Combinatorics
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In this paper we study the diameter of the random graph G(n, p), i.e., the largest finite distance between two vertices, for a wide range of functions p = p(n). For p = λ/n with λ 1 constant we give a simple proof of an essentially best possible result, with an Op(1) additive correction term. Using similar techniques, we establish two-point concentration in the case that np → ∞. For p =(1 + ε)/n with ε → 0, we obtain a corresponding result that applies all the way down to the scaling window of the phase transition, with an Op(1/ε) additive correction term whose (appropriately scaled) limiting distribution we describe. Combined with earlier results, our new results complete the determination of the diameter of the random graph G(n, p) to an accuracy of the order of its standard deviation (or better), for all functions p = p(n). Throughout we use branching process methods, rather than the more common approach of separate analysis of the 2-core and the trees attached to it.