Problem stopnje in premera

Matjašič, Telopea

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Title:	Problem stopnje in premera
Authors:	ID Matjašič, Telopea (Author) ID Jakovac, Marko (Mentor) More about this mentor...
Files:	UN_Matjasic_Telopea_2016.pdf (502,18 KB) MD5: A693851AD011057CE6FC695B986EC8D7
Language:	Slovenian
Work type:	Undergraduate thesis
Typology:	2.11 - Undergraduate Thesis
Organization:	FNM - Faculty of Natural Sciences and Mathematics
Abstract:	V diplomskem delu je obravnavan problem stopnje in premera. Poiskati hočemo največji graf, glede na število njegovih vozlišč, ki bo imel premer k ≥ 1 in največjo stopnjo vozlišč d ≥ 1. Število vozlišč takšnega grafa je navzgor omejeno z Mooreovo mejo: 1 + d sum_{i=0}^{k-1}(d-1)^i. Graf, ki doseže to mejo, imenujemo Mooreov graf. Izkaže se, da je zelo malo grafov, ki dosežejo to mejo, zato se je smiselno osredotočiti na grafe, katerih število vozlišč je blizu tej meji. Iskanje teh grafov poteka na dva načina. Najprej pogledamo kako so z dokazi o neobstoju grafov zniževali zgornjo mejo, nato pa predstavimo nekatere splošne metode s katerimi so izboljševali spodnjo mejo in se na ta način približali Mooreovi meji. Ugotovimo, da je do sedaj znanih le malo grafov, ki so optimalne velikosti oziroma, ki odgovorijo na naš zastavljen problem. Posebej opišemo grafe, za katere je premer bodisi 2 bodisi 3.
Keywords:	stopnja, premer, Mooreov graf, Mooreova meja.
Place of publishing:	Maribor
Publisher:	[T. Matjašič]
Year of publishing:	2016
PID:	20.500.12556/DKUM-60295
UDC:	519.17(043.2)
COBISS.SI-ID:	22476552
NUK URN:	URN:SI:UM:DK:JVF5QPBQ
Publication date in DKUM:	30.08.2016
Views:	1246
Downloads:	99
Metadata:
Categories:	FNM
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Secondary language

Language:	English
Title:	The degree diameter problem
Abstract:	This graduate thesis addresses the degree diameter problem. We want to find the largest graph, with respect to its number of vertices, with diameter k ≥ 1 and maximum degree d ≥ 1. The number of vertices of such graph is bounded above by the Moore bound: 1 + d sum_{i=0}^{k-1}(d-1)^i. A graph that reaches this bound is called a Moore graph. It turns out that there are only a small number of graphs reaching this bound, so it has more sense to focus on graphs whose order is close to this bound. There are two ways of finding such graphs. First, we look how the upper bound was lowered by proving the non-existence of such graphs, and then we present some general methods that where used to improve the lower bound. We show that there are only a few graphs known to be optimal and correspond to our particular problem. The graphs with diameter 2 or 3, respectively, are also described.
Keywords:	degree, diameter, Moore graph, Moore bound.

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