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Title: Some applications of graph theory
Author: Eggemann, Nicole
ISNI:       0000 0004 2725 4174
Awarding Body: Brunel University
Current Institution: Brunel University
Date of Award: 2009
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We investigate four different applications on graph theory. First we show for a generalisation of the well-known Barab´asi-Albert model that the expectation of the clustering coefficient of this graph process is asymptotically proportional to log n/n,, by generalising a result of Bollob´as and Riordan. Secondly, we investigate the complexity of searching for a given vertex in a scale-free graph, using only locally gathered information. We consider two kinds of models which are generalisations of the Barab´asi-Albert model, proving two lower bounds of Ω(n1/2) on the expected time to find the worstcase target, under a restrictive model of local information. Thirdly, we consider two orientation problems in a graph, namely the minimisation of the sum of shortest paths lengths and the minimisation of the diameter. We show that it is NP-complete to determine whether a graph has an orientation for which the sum of shortest paths lengths is less than an integer specified in the input. Furthermore we describe an algorithm that runs in linear time and decides for a planar graph G whether there is an orientation such that the diameter of G is less than a fixed constant. Fourthly, we consider the well-known k-L(2, 1)-labelling which is a mapping from the vertex set of a graph G = (V, E) into an interval of integers {0, . . . , k} such that any two adjacent vertices are mapped onto integers that are at least two apart, and every two vertices with a common neighbour are mapped onto distinct integers. We show that the k-L(2, 1)-labelling is NP-complete for planar graphs and any k ≥ 4 by reduction from Planar Cubic Two-Colourable Perfect Matching. Schaefer stated without proof that Planar Cubic Two-Colourable Perfect Matching is NP-complete. In this thesis we give a proof of this.
Supervisor: Noble, S.; Rodgers, G. J. Sponsor: Marie Curie Early Stage Training Fellowship (NET-ACE-programme) under grant number MEST-CT-2004-6724
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available