Preštevanje dreves

diplomsko delo

Jera Stojko (Author), Boštjan Kuzman (Mentor)

Abstract

V teoriji grafov drevo pomeni kombinatoričen objekt, ki ga običajno definiramo kot povezan graf brez ciklov. V nalogi je predstavljen problem preštevanja različnih dreves s podanim številom vozlišč. Podani so štirje dokazi znamenitega Cayleyevega izreka za število označenih dreves: dokaz s Prüferjevo bijektivno konstrukcijo, dokaz s preštevanjem zakoreninjenih dreves na dva načina J. Pitmana, dokaz z rekurzivno formulo za število označenih gozdov in dokaz z uporabo Kirchhoffove zveze med determinantami matrik in vpetimi drevesi. V zadnjem delu je izpeljan novejši rezultat A. China in ostalih, o verjetnosti, da je naključno izbrano drevo v polnem grafu vpeto drevo. Posebej je obravnavano obnašanje limitne vrednosti verjetnosti, ko število vozlišč polnega grafa raste čez vse meje. Dobljeni rezultat je presenetljiv in lep.

Keywords

graf;polni graf;drevo;vpeto drevo;preštevanje;Cayleyev izrek;verjetnost;faktor;

Data

Language:	Slovenian
Year of publishing:	2017
Typology:	2.11 - Undergraduate Thesis
Organization:	UL PEF - Faculty of Education
Publisher:	[J. Stojko]
UDC:	519.17(043.2)
COBISS:	11727689
Views:	810
Downloads:	185
Average score:	0 (0 votes)
Metadata:

Other data

Secondary language:	English
Secondary title:	Counting trees
Secondary abstract:	In graph theory, trees are combinatorial objects usually defined as connected graphs without cycles. In this thesis, the problem of counting different trees with a given number of vertices is presented. Four proofs of a celebrated theorem on the number of labeled trees by A. Cayley are given: by using a bijective construction due to H. Prüfer, by double counting of labeled rooted trees due to J. Pitman, by establishing a recursion on the number of labeled forests, and finally, by applying a relation between determinants and spanning trees due to H. Kirchhoff. In the final part, a recent result by A. Chin et al. is presented on the probability that a randomly picked tree in a complete graph is a spanning tree. In particular, the limit of this value as the number of vertices approaches infinity is observed. The result obtained is both surprising and beautiful.
Secondary keywords:	mathematics;matematika;
File type:	application/pdf
Type (COBISS):	Bachelor thesis/paper
Thesis comment:	Univ. v Ljubljani, Pedagoška fak., Dvopredmetni učitelj: Fizika-matematika
Pages:	[43] str.
ID:	10865631