Loading...

Tournaments

Shahsavaran, Mohsen | 2014

861 Viewed
  1. Type of Document: M.Sc. Thesis
  2. Language: Farsi
  3. Document No: 46409 (02)
  4. University: Sharif University of Technology
  5. Department: Mathematical Science
  6. Advisor(s): Mahmoodian, Ebadollah; Moghadasi, Reza
  7. Abstract:
  8. In this thesis which is an extensive expository and survey, we study concepts like irreducible, primitive, regular and almost regularournaments. We study and show the proof of some graph theoretical properties of tournaments such as being pancyclic, vertex or arc-pancyclic, and a lower bound on the number of cycles of each length in irreducible tournaments. Spectral properties and the Perron eigenvalue of tournament matrices are also extensively discussed.The ranks of tournament matrices over different algebraic fields, is another subjectthat is covered. We have written computer programs to study the distribution of the rank of tournament matrices over Z2.With a little more work,one might dare to make a conjecture on this issue.As an attempt to try our chance in proving Maybee and Pullman’s Conjecture on the asymptotic probability of invertibility of tournament matrices over the field of real numbers, we study two already proved related subjects, namely the asymptotic probability of invertibility of (0; 1) matrices and that of symmetric (0;1)matrices. This study introduces the powerful and potentially useful tool of Littlewood-Offord theory, both in linear and quadratic forms. Finally we study some properties of the well-known Brualdi-Li matrix and discuss the proof of the Brualdi-Li Conjecture in detail
  9. Keywords:
  10. Perron-Frobenious Theorem ; Tournament ; Irreducible Tournament ; Tournament Matrix Rank ; Littlewood- Offord Problem ; Brualdi-Li Conjecture

 Digital Object List

 Bookmark

...see more