Properties of the Brualdi–Li tournament matrix

Abstract

The Brualdi–Li tournament matrix is conjectured to have the largest spectral radius among all tournament matrices of even order. In this paper two forms of the characteristic polynomial of the Brualdi–Li tournament matrix are found. Using the first form it is shown that the roots of the characteristic polynomial are simple and that the Brualdi–Li tournament matrix is diagonalizable. Using the second form an expression is found for the coefficients of the powers of the variable λ in the characteristic polynomial. These coefficients give information about the cycle structure of the cycles of length 1–5 of the directed graph associated with the Brualdi–Li tournament matrix.