Chapter 45

Tournaments

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 ...

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