Two investigations on tournaments: intersection spectra and score sequences

This thesis focuses on two ideas in tournament theory: cycle intersections in tournaments (i.e., intersection spectra of tournaments) and some work done on score sequences of tournaments. Chapter 2 deals with the intersection spectrum of a specific family of tournaments, and Chapter 4 restricts the intersection spectrum of strong tournaments to characterize new families of strong tournaments. Previous work by Brualdi and Li, as well as, Poet and Shader, characterizes what are known as upset tournaments by their score sequences. Chapter 3 extends this characterization to k-upset tournaments, again based on their score sequences (in particular, their score arrangements). Chapter 5 provides a new proof of a well-known characterization of score sequences of tournaments. At the very end of this thesis is an Appendix that contains many drawings of tournaments referred to in the body of this work. It also contains the Graph Theory Hymn as a special tribute to Bohdan Zelinka whose work on cyclically simple tournaments gave the inspiration for this thesis. Keywords: tournaments, upset tournaments, intersection spectrum, score sequences