The height of random k‐trees and related branching processes1. Issue 4 (6th February 2015) - Cooper, Colin; Frieze, Alan; Uehara, Ryuhei
Abstract: We consider the height of random k ‐trees and k ‐Apollonian networks. These random graphs are not really trees, but instead have a tree‐like structure. The height will be the maximum distance of a vertex from the root. We show that w.h.p. the height of random k ‐trees and k ‐Apollonian networks is asymptotic to c log t, where t is the number of vertices, and c = c ( k ) is given as the solution to a transcendental equation. The equations are slightly different for the two types of process. In the limit as k → ∞ the height of both processes is asymptotic to log t / ( k log 2 ) . © 2014 Wiley Periodicals, Inc. Random Struct. Alg., 45, 675–702, 2014