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Analysis of tree spectra
Thesis (PhD)--Stellenbosch University, 2018.
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| Format: | Thesis |
| Language: | en_ZA |
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Stellenbosch : Stellenbosch University
2018
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| _version_ | 1867613998401191936 |
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| access_status_str | Open Access |
| author | Dadedzi, Kenneth |
| author2 | Wagner, Stephan |
| author_browse | Dadedzi, Kenneth Wagner, Stephan |
| author_facet | Wagner, Stephan Dadedzi, Kenneth |
| author_sort | Dadedzi, Kenneth |
| collection | Thesis |
| dc_rights_str_mv | Stellenbosch University |
| description | Thesis (PhD)--Stellenbosch University, 2018. |
| format | Thesis |
| id | oai:scholar.sun.ac.za:10019.1/104848 |
| institution | Stellenbosch University (South Africa) |
| language | en_ZA |
| last_indexed | 2026-06-10T12:45:01.662Z |
| license_str | Other — see source repository |
| provenance_str_mv | Harvested via OAI-PMH from SUNScholar — Stellenbosch University Repository |
| publishDate | 2018 |
| publishDateRange | 2018 |
| publishDateSort | 2018 |
| publisher | Stellenbosch : Stellenbosch University |
| publisherStr | Stellenbosch : Stellenbosch University |
| record_format | dspace |
| source_str | SUNScholar — Stellenbosch University Repository |
| spelling | oai:scholar.sun.ac.za:10019.1/104848 Analysis of tree spectra Dadedzi, Kenneth Wagner, Stephan Stellenbosch University. Faculty of Science. Dept. of Mathematical Sciences. Division Mathematics. Binary system (Mathematics) Eigenvalues Laplacian matrices Trees (Graph theory) UCTD Tree spectra Thesis (PhD)--Stellenbosch University, 2018. ENGLISH ABSTRACT : We study the set of eigenvalues (spectrum) of the adjacency matrix, Laplacian matrix and the distance matrix of trees. In particular, we focus on the distribution of eigenvalues in the spectra of large random trees. The families of random trees considered in this work are simply generated trees and increasing trees. We prove that attaching several copies (two or more) of a tree H to vertices in a tree T “forces" certain real numbers into the adjacency, Laplacian or distance spectrum of the resulting tree. With this construction of forcing subtrees, we prove that the mean proportion of an eigenvalue α in the spectrum of a large random tree is at least the mean number of occurrences of a specific forcing subtree in the large random tree. This gives us explicit lower bounds on the asymptotic mean multiplicity of eigenvalues for different families of random trees. We prove that the mean proportion of an eigenvalue α in the spectrum of the adjacency matrix and Laplacian matrix of a large simply generated tree can be obtained by solving a system of functional equations. We provide an algorithm to solve this system numerically for a given eigenvalue. For instance, using this algorithm, we show that on average approximately 1.4%, 2.1%, 2.5% and 3.3% of the spectrum of a large pruned binary tree, labelled rooted tree, plane tree and pruned ternary tree respectively consist of the eigenvalue 1. Further, we provide explicit formulas for computing the mean proportion of the eigenvalue 0 in the spectrum of a large simply generated tree. We also study the spectra of recursive trees and binary increasing trees. We show that the distribution of the eigenvalue 0 (and other eigenvalues) in these random trees satisfies a central limit theorem. We also compute the mean and variance of the multiplicity of the eigenvalue 0 in the spectrum of a large recursive tree and binary increasing tree. The final chapter deals with related, but somewhat different questions. Given a rooted tree T with leaves v1, v2, . . . , vn, we define the ancestral matrix C(T) of T to be the n × n matrix for which the entry in the i-th row, j-th column is the level (distance from the root) of the first common ancestor of vi and vj . We study properties of this matrix, in particular regarding its spectrum: we obtain several upper and lower bounds for the eigenvalues in terms of other tree parameters. We also find a combinatorial interpretation for the coefficients of the characteristic polynomial of C(T), and show that for dary trees, a specific value of the characteristic polynomial is independent of the precise shape of the tree. AFRIKAANSE OPSOMMING : Ons bestudeer die versameling van eiewaardes (spektrum) van die nodusmatriks, Laplace se matriks en die afstandmatriks van bome. In die besonder fokus ons op die verdeling van eiewaardes in die spektra van groot vir ’n gegewe eiewaarde. Byvoorbeeld kan ons met behulp van hierdie algoritme wys dat ’n gemiddeld van 1.4%, 2.1%, 2.5% en 3.3% van die spektrum van ’n groot gesnoeide binêre boom, gemerkte wortelboom, vlak boom en gesnoeide ternêre boom onderskeidelik uit die eiewaarde 1 bestaan. Verder gee ons eksplisiete formules vir die berekening van die gemiddelde proporsie van die eiewaarde 0 in die spektrum van ’n groot eenvoudig gegenereerde boom. Ons bestudeer ook die spektra van rekursiewe bome en binêre toenemende bome. Ons wys dat die verdeling van die eiewaarde 0 (en ander eiewaardes) in hierdie lukrake bome ’n sentrale limietstelling bevredig. Ons bereken ook die gemiddeld en die variansie van die veelvoudigheid van die eiewaarde 0 in die spektrum van ’n groot rekursiewe boom en binêre toenemende boom. Die laaste hoofstuk handel oor verwante, maar ietwat ander vrae. Gegewe ’n gewortelde boom T met blare v1, v2, . . . , vn, definieer ons die voorouermatriks C(T) van T as die n × n matriks waarvoor die inskrywing in die ide ry, j-de kolom die vlak (afstand vanaf die wortel) van die eerste gemene voorouer van vi en vj is. Ons bestudeer eienskappe van hierdie matriks, veral ten opsigte van sy spektrum: ons kry verskeie bo- en ondergrense vir die eiewaardes in terme van ander boomparameters. Ons vind ook ’n kombinatoriese interpretasie vir die koëffisiënte van die karakteristieke polinoom van C(T), en toon aan dat vir d-êre bome ’n spesifieke waarde van die karakteristieke polinoom onafhanklik is van die presiese vorm van die boom. lukrake bome. Die families van lukrake bome wat in hierdie werk beskou word, is eenvoudig gegenereerde bome en toenemende bome. Ons bewys dat ons ’n aantal kopieë (twee of meer) van ’n boom H aan noduse van ’n boom T kan aanheg om sekere reële getalle in die spektra van die nodusmatriks, Laplace se matriks en die afstandmatriks van die boom wat ontstaan te “forseer”. Met hierdie konstruksie van forserende deelbome bewys ons dat die gemiddelde proporsie van ’n eiewaarde α in die spektrum van ’n groot lukrake boom minstens die gemiddelde aantal kopieë van ’n spesifieke forserende deelboom in die groot lukrake boom is. Dit lewer eksplisiete ondergrense vir die asimptotiese gemiddelde veelvoudigheid van eiewaardes vir verskillende families van lukrake bome. Ons bewys dat die gemiddelde proporsie van ’n eiewaarde α in die spektrum van die nodusmatriks en Laplace se matriks van ’n groot eenvoudig gegenereerde boom verkry kan word deur ’n stelsel funksionele vergelykings op te los. Ons gee ’n algoritme om hierdie stelsel numeries op te los Doctoral 2018-10-09T12:33:14Z 2018-12-07T06:47:42Z 2018-10-09T12:33:14Z 2018-12-07T06:47:42Z 2018-12 Thesis http://hdl.handle.net/10019.1/104848 en_ZA Stellenbosch University x, 117 pages : illustrations application/pdf Stellenbosch : Stellenbosch University |
| spellingShingle | Binary system (Mathematics) Eigenvalues Laplacian matrices Trees (Graph theory) UCTD Tree spectra Dadedzi, Kenneth Analysis of tree spectra |
| title | Analysis of tree spectra |
| title_full | Analysis of tree spectra |
| title_fullStr | Analysis of tree spectra |
| title_full_unstemmed | Analysis of tree spectra |
| title_short | Analysis of tree spectra |
| title_sort | analysis of tree spectra |
| topic | Binary system (Mathematics) Eigenvalues Laplacian matrices Trees (Graph theory) UCTD Tree spectra |
| url | http://hdl.handle.net/10019.1/104848 |
| work_keys_str_mv | AT dadedzikenneth analysisoftreespectra |