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- Type of Document: M.Sc. Thesis
- Language: Farsi
- Document No: 47572 (02)
- University: Sharif University of Technology
- Department: Mathematical Sciences
- Advisor(s): Mahmoudian, Ebadollah
- Abstract:
- In 1953, Landau expressed the conditions for the existence of tournaments with a prescribed score vector R. Eleven years later, Ryser proposed an algorithm for constructing the tournament matrix with score vector R and showed that any two tournaments with the same score vector can be obtained from one another by a sequence of 3-cycle switches. Last year (2014) Brualdi and Fristcher explained in an article, similar to this process, the existence of, algorithm and switches for loopy tournament and Hankel tournament and similar structure skew-Hankel tournament. This paper is discussed in a chapter of this thesis. In 1991, De Caen found a lower bound for the real rank of a tournament matrix. This result led to have a better bounds for other ranks of tournament matrix including the Boolean rank, nonnegative rank, and term rank. The following topics are discussed in another chapter of this thesis. Relationships between the tournament ranks, introducing classes of tournament matrices which satisfy various inequality relationship between the tournament matrix ranks, identification of tournaments with known ranks, a relationship between score vector and different ranks, a bipartite graph corresponding to a tournament and the impact of changing the direction of arcs, on tournament ranks. We will look at briefly to 2-tournaments and in all of the existential theorems, the critical case method is our preferred method of proof in this discussion
- Keywords:
- Hankel Tournament ; Tournament ; Skew-Hankel Tournament ; Boolean Rank ; Term Rank ; Real Rank ; Nonnegative Rank ; Critical Case Method
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