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Connectivity Properties of Archimedean and Laves LatticesPrimeFaces.cw("AccordionPanel","widget_formSmash_some",{id:"formSmash:some",widgetVar:"widget_formSmash_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_all",{id:"formSmash:all",widgetVar:"widget_formSmash_all",multiple:true});
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2004 (English)Doctoral thesis, comprehensive summary (Other academic)
##### Abstract [en]

##### 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

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt496",{id:"formSmash:j_idt496",widgetVar:"widget_formSmash_j_idt496",multiple:true});
##### Supervisors

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt503",{id:"formSmash:j_idt503",widgetVar:"widget_formSmash_j_idt503",multiple:true});
#####

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt510",{id:"formSmash:j_idt510",widgetVar:"widget_formSmash_j_idt510",multiple:true}); Available from: 2004-05-14 Created: 2004-05-14Bibliographically approved
##### List of papers

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.

1. Inclusions and Non-Inclusions of Archimedean and Laves Lattices$(function(){PrimeFaces.cw("OverlayPanel","overlay164540",{id:"formSmash:j_idt564:0:j_idt568",widgetVar:"overlay164540",target:"formSmash:j_idt564:0:partsLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

2. Ordering Bond Percolation Critical Probabilities$(function(){PrimeFaces.cw("OverlayPanel","overlay164541",{id:"formSmash:j_idt564:1:j_idt568",widgetVar:"overlay164541",target:"formSmash:j_idt564:1:partsLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

3. Bounds for the Connective Constant on the Hexagonal Lattice$(function(){PrimeFaces.cw("OverlayPanel","overlay164542",{id:"formSmash:j_idt564:2:j_idt568",widgetVar:"overlay164542",target:"formSmash:j_idt564:2:partsLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

4. Pairs of Graphs with Connective Constants and Critical Probabilities in the Same Order$(function(){PrimeFaces.cw("OverlayPanel","overlay164543",{id:"formSmash:j_idt564:3:j_idt568",widgetVar:"overlay164543",target:"formSmash:j_idt564:3:partsLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

5. Inclusions and Non-Inclusions of Matching Archimedean Lattices$(function(){PrimeFaces.cw("OverlayPanel","overlay164544",{id:"formSmash:j_idt564:4:j_idt568",widgetVar:"overlay164544",target:"formSmash:j_idt564:4:partsLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

6. Estimation of Bond Percolation Thresholds on the Archimedean Lattices$(function(){PrimeFaces.cw("OverlayPanel","overlay164545",{id:"formSmash:j_idt564:5:j_idt568",widgetVar:"overlay164545",target:"formSmash:j_idt564:5:partsLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

isbn
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