The height of random k‐trees and related branching processes1. Issue 4 (21st October 2014)

• Record Type: Journal Article
• Authors: Cooper, Colin; Frieze, Alan; Uehara, Ryuhei
• Is Part Of: Random structures & algorithms. Volume 45:Issue 4(2014)
• Page Start: 675
• Page End: 702
• Publication Date: 2014-10-21
• Subjects: Random graphs -- Periodicals; Mathematical analysis -- Periodicals; 519
• DOI: 10.1002/rsa.20576
• Languages: English
• ISSNs: 1042-9832

Abstract

<abstract abstract-type="main"> <title>Abstract</title> <p>We consider the height of random <italic>k</italic>‐trees and <italic>k</italic>‐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 <italic>k</italic>‐trees and <italic>k</italic>‐Apollonian networks is asymptotic to <italic>c</italic> log <italic>t</italic>, where <italic>t</italic> is the number of vertices, and <italic>c</italic> = <italic>c</italic>(<italic>k</italic>) is given as the solution to a transcendental equation. The equations are slightly different for the two types of process. In the limit as <italic>k</italic> →<italic>∞</italic> the height of both processes is asymptotic to <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgh21kb0xms" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:10429832:media:rsa20576:rsa20576-math-0001" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>log</mml:mi><mml:mi>t</mml:mi><mml:mo>/</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mi>k</mml:mi><mml:mi>log</mml:mi><mml:mn>2</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>. © 2014 Wiley Periodicals, Inc. Random Struct. Alg., 45, 675–702, 2014</p> </abstract>