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Obtaining Polynomial Invariants for Rooted Trees from their Random Destruction
(Engelska)Ingår i: Artikel i tidskrift (Övrigt vetenskapligt) Submitted
Nationell ämneskategori
Diskret matematik Sannolikhetsteori och statistik
Identifikatorer
URN: urn:nbn:se:uu:diva-500976OAI: oai:DiVA.org:uu-500976DiVA, id: diva2:1753792
Tillgänglig från: 2023-04-28 Skapad: 2023-04-28 Senast uppdaterad: 2023-04-29
Ingår i avhandling
1. Building and Destroying Urns, Graphs, and Trees
Öppna denna publikation i ny flik eller fönster >>Building and Destroying Urns, Graphs, and Trees
2023 (Engelska)Doktorsavhandling, sammanläggning (Övrigt vetenskapligt)
Abstract [en]

In this thesis, consisting of an introduction and four papers, different models in the mathematical area of combinatorial probability are investigated.

In Paper I, two operations for combining generalised Pólya urns, called disjoint union and product, are defined. This is then shown to turn the set of isomorphism classes of Pólya urns into a semiring, and we find that assigning to an urn its intensity matrix is a semiring homomorphism.

In paper II, a modification and generalisation of the random cutting model is introduced. For a finite graph with given source and target vertices, we remove vertices at random and discard all resulting components without a source node. The results concern the number of cuts needed to remove all target vertices and the size of the remaining graph, and suggest that this model interpolates between the traditional cutting model and site percolation.

In paper III, we define several polynomial invariants for rooted trees based on the modified cutting model in Paper II.We find recursive identities for these invariants and, using an approach via irreducibility of polynomials, prove that two specific invariants are complete, that is, they distinguish rooted trees up to isomorphism.

In paper IV, joint with Paul Thévenin, we consider an operation of concatenating t random perfect matchings on 2n vertices. Our analysis of the resulting random graph as t tends to infinity shows that there is a giant component if and only if n is odd, and that the size of this giant component as well as the number of components is asymptotically normally distributed.

Ort, förlag, år, upplaga, sidor
Uppsala: Department of Mathematics, 2023. s. 33
Serie
Uppsala Dissertations in Mathematics, ISSN 1401-2049 ; 131
Nationell ämneskategori
Sannolikhetsteori och statistik Diskret matematik
Identifikatorer
urn:nbn:se:uu:diva-500978 (URN)978-91-506-3009-1 (ISBN)
Disputation
2023-08-24, Häggsalen, Ångströmlaboratoriet, Lägerhyddsvägen 1, Uppsala, 13:15 (Engelska)
Opponent
Handledare
Tillgänglig från: 2023-05-29 Skapad: 2023-04-28 Senast uppdaterad: 2023-05-30

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Burghart, Fabian
Diskret matematikSannolikhetsteori och statistik

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