Properties of the Brualdi–Li tournament matrix

Some properties complementary to Brualdi-Li matrices

“…One can easily show that equation (3.1) has 2n distinct roots and among them there are exactly one real positive root and one real negative real root. Therefore, Theorem 3.2 implies that B 2n has 2n distinct eigenvalues, generating an alternative proof of Theorem 3 of [9].…”

“…In [9] Kirkland derived several eigen-properties of Brualdi-Li matrix B 2n and showed that every Brualdi-Li matrix has distinct eigenvalues and so it is diagonalizable. In this section we derive further properties of this matrix and, as a by-product, reprove that every Brualdi-Li matrix has distinct eigenvalues.…”

“…Because B 2n is known to have distinct eigenvalues [9], it suffices to show that B 2n has exactly one real negative eigenvalue λ 2 . From Theorem 5 of [9], the characteristic polynomial c(λ) of B 2n is given by…”

“…Tournament matrices have been well studied in the past decades (see [2], [6], [7], [9], [23] and their references). A wealth of literature focuses on deriving algebraic or combinatorial attributes of the matrices because of their interplays between matrix/graph theoretic and spectral properties [1], [2], [6], [16], [20], [23].…”

On 5‐Cycles and 6‐Cycles in Regular n‐Tournaments

“…Hence, summing from 1 to n in formula (16) and then dividing by 10 yield an identity relating c 4 (T ) and c 5 (T ).…”