T. S. Michael
United States Naval Academy
A tournament is a directed graph whose adjacency matrix A satisfies the matrix equation A + AT + I = J, where J is a matrix of 1’s. The matrix equation is the key to the interplay between the graph-theoretic structure of tournaments and properties of their adjacency matrices. The classic book by Moon [Moo68] and the more recent surveys by Reid and Beineke [RB78] and Reid [Rei04] emphasize graph-theoretic aspects of tournaments. In this chapter, we emphasize properties of tournaments related to linear algebra and combinatorial matrix theory.
45.1 Tournaments and Tournament Matrices
Suppose that a round-robin tournament is held among a set of players. Each player meets each other ...