The height of random k‐trees and related branching processes1. Issue 4 (21st October 2014) - Cooper, Colin
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 logt/(klog2). © 2014 Wiley Periodicals, Inc. Random Struct. Alg., 45, 675–702, 2014