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Connectivity Properties of Archimedean and Laves Lattices
Uppsala University, Teknisk-naturvetenskapliga vetenskapsområdet, Mathematics and Computer Science, Department of Mathematics.
2004 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

An Archimedean lattice is a graph of a regular tiling of the plane, such that all corners are equivalent. A tiling is regular if all tiles are regular polygons: equilateral triangles, squares, et cetera. There exist exactly 11 Archimedean lattices. Being planar graphs, the Archimedean lattices have duals, 3 of which are Archimedean, the other 8 are called Laves lattices.

In the thesis, three measures of connectivity of these 19 graphs are studied: the connective constant for self-avoiding walks, and bond and site percolation critical probabilities. The connective constant measures connectivity by the number of walks in which all visited vertices are unique. The critical probabilities quantify the proportion of edges or vertices that can be removed, so that the produced subgraph has a large connected component.

A common issue for these measures is that they, although intensely studied by both mathematicians and scientists from other fields, have been calculated only for very few graphs. With the goal of comparing the induced orders of the Archimedean and Laves lattices under the three measures, the thesis gives improved bounds and estimates for many graphs.

A large part of the thesis focuses on the problem of deciding whether a given graph is a subgraph of another graph. This, surprisingly difficult problem, is considered for the set of Archimedean and Laves lattices, and for the set of matching Archimedean and Laves lattices.

Place, publisher, year, edition, pages
Uppsala: Matematiska institutionen , 2004. , p. 37
Series
Uppsala Dissertations in Mathematics, ISSN 1401-2049 ; 34
Keywords [en]
Mathematical statistics, percolation, self-avoiding walks, Archimedean and Laves lattices
Keywords [sv]
Matematisk statistik
National Category
Probability Theory and Statistics
Identifiers
URN: urn:nbn:se:uu:diva-4251ISBN: 91-506-1751-6 (print)OAI: oai:DiVA.org:uu-4251DiVA, id: diva2:164546
Public defence
2004-06-04, Room 2347, Polacksbacken, Uppsala, 13:15
Opponent
Supervisors
Available from: 2004-05-14 Created: 2004-05-14Bibliographically approved
List of papers
1. Inclusions and Non-Inclusions of Archimedean and Laves Lattices
Open this publication in new window or tab >>Inclusions and Non-Inclusions of Archimedean and Laves Lattices
Article in journal (Refereed) Submitted
Identifiers
urn:nbn:se:uu:diva-91714 (URN)
Available from: 2004-05-14 Created: 2004-05-14Bibliographically approved
2. Ordering Bond Percolation Critical Probabilities
Open this publication in new window or tab >>Ordering Bond Percolation Critical Probabilities
Article in journal (Refereed) Submitted
Identifiers
urn:nbn:se:uu:diva-91715 (URN)
Available from: 2004-05-14 Created: 2004-05-14Bibliographically approved
3. Bounds for the Connective Constant on the Hexagonal Lattice
Open this publication in new window or tab >>Bounds for the Connective Constant on the Hexagonal Lattice
2004 In: Journal of Physics A: Mathematical and General, Vol. 3, no 37, p. 549-560Article in journal (Refereed) Published
Identifiers
urn:nbn:se:uu:diva-91716 (URN)
Available from: 2004-05-14 Created: 2004-05-14Bibliographically approved
4. Pairs of Graphs with Connective Constants and Critical Probabilities in the Same Order
Open this publication in new window or tab >>Pairs of Graphs with Connective Constants and Critical Probabilities in the Same Order
Article in journal (Refereed) Submitted
Identifiers
urn:nbn:se:uu:diva-91717 (URN)
Available from: 2004-05-14 Created: 2004-05-14Bibliographically approved
5. Inclusions and Non-Inclusions of Matching Archimedean Lattices
Open this publication in new window or tab >>Inclusions and Non-Inclusions of Matching Archimedean Lattices
Article in journal (Refereed) Submitted
Identifiers
urn:nbn:se:uu:diva-91718 (URN)
Available from: 2004-05-14 Created: 2004-05-14Bibliographically approved
6. Estimation of Bond Percolation Thresholds on the Archimedean Lattices
Open this publication in new window or tab >>Estimation of Bond Percolation Thresholds on the Archimedean Lattices
Article in journal (Refereed) Submitted
Identifiers
urn:nbn:se:uu:diva-91719 (URN)
Available from: 2004-05-14 Created: 2004-05-14Bibliographically approved

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