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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.... 

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... 

In this thesis, using linear programming, we present approximation algorithms for two groups of related problems, all of which are better than previously known algorithms in terms of approximation factor or running time. The first problem is the minimum feedback arc set on tournaments, a well-known NP-hard problem. In this problem, a tournament (complete directed weighted graph) is given as input, and the goal is to find a minimum-sized subset of its edges, whose removal makes the tournament acyclic. We will present a randomized 2.127-approximation algorithm based on the standard linear programming of this problem, for tournaments that satisfy probability constraints. We also introduce a...