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Eplett's theorem for self-converse generalised tournaments
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.ORCID iD: 0000-0002-0592-1808
2018 (English)In: The Australasian Journal of Combinatorics, ISSN 1034-4942, Vol. 70, no 3, p. 329-335Article in journal (Refereed) Published
Abstract [en]

The converse of a tournament is obtained by reversing all arcs. If a tournament is isomorphic to its converse, it is called self-converse. Eplett provided a necessary and sufficient condition for a sequence of integers to be realisable as the score sequence of a self-converse tournament. In this paper we extend this result to generalised tournaments.

Place, publisher, year, edition, pages
2018. Vol. 70, no 3, p. 329-335
National Category
Mathematics
Identifiers
URN: urn:nbn:se:uu:diva-335318ISI: 000424085800004OAI: oai:DiVA.org:uu-335318DiVA, id: diva2:1162325
Available from: 2017-12-04 Created: 2017-12-04 Last updated: 2018-03-28Bibliographically approved
In thesis
1. Degrees in Random Graphs and Tournament Limits
Open this publication in new window or tab >>Degrees in Random Graphs and Tournament Limits
2018 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

This thesis consists of an introduction and six papers on the topics of degree distributions in random graphs and tournaments and their limits.

The first two papers deal with a dynamic random graph, evolving in time through duplication and deletion of vertices and edges. In Paper I we study the degree densities of this model. We show that these densities converge almost surely and determine their limiting values exactly as well as asymptotically for large degrees. In Paper II we study the evolution of the maximum degree and provide a precise growth rate thereof.

Paper III deals with a dynamic random tree model known as the vertex-splitting tree model. We show that the degree densities converge almost surely and find an infinite linear system of equations which they must satisfy. Unfortunately we are not able to show that this system has a unique solution except in special cases.

Paper IV is about self-converse generalised tournaments. A self-converse generalised tournament can be seen as a matrix whose entries take values in [0,1] and whose diagonally opposite elements sum to 1. We characterise completely the marginals of such a matrix, and show that such marginals can always be realised by a self-converse generalised tournament.

In Paper V, we define and develop the theory of tournament limits and tournament kernels. We characterise transitive and irreducible tournament limits and kernels, and prove that any tournament limit and kernel has an essentially unique decomposition into irreducible tournament limits or kernels interlaced by a transitive part.

In Paper VI, we study the degree distributions of tournament limits, or equivalently, the marginals of tournament kernels. We describe precisely which distributions on [0,1] which may appear as degree distributions of tournament limits and which functions from [0,1] to [0,1] may appear as the marginals of tournament kernels. Moreover, we show that any distribution or marginal on this form may be realised by a tournament limit or tournament kernel. We also study those distributions and marginals which can be realised by a unique tournament limit or kernel, and find that only the transitive tournament limit/kernel gives rise to a degree distribution or marginal with this property.

Place, publisher, year, edition, pages
Uppsala: Department of Mathematics, 2018. p. 26
Series
Uppsala Dissertations in Mathematics, ISSN 1401-2049 ; 105
Keywords
Random graphs, degree distributions, degree sequences, graph limits, tournaments
National Category
Mathematics
Research subject
Mathematics
Identifiers
urn:nbn:se:uu:diva-339025 (URN)978-91-506-2677-3 (ISBN)
Public defence
2018-03-09, Polhemssalen, Ångströmlaboratoriet, Lägerhyddsvägen 1, Uppsala, 13:15 (English)
Opponent
Supervisors
Available from: 2018-02-14 Created: 2018-01-17 Last updated: 2018-02-14

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