Abstract
LetB2ndenote the Brualdi-Li matrix, and letρ2n=ρ(B2n)denote the Perron value of the Brualdi-Li matrix of order2n. We prove that2n(n-1/2-ρ2n)is monotonically decreasing for allnandρ2n<n-1/2-(e2-1)/4(e2+1)n, wheree=2.718281828459045….
Highlights
A tournament matrix of order n is a (0, 1) matrix T satisfying the equation T + Tt = J − I, where J is the all ones matrix, I is the identity matrix, and Tt is the transpose of T
The matrix B2n has been dubbed by the Brualdi-Li matrix
Kirkland [6] conjectured that 2n(n − 1/2 − ρ2n) is monotonically decreasing for all n
Summary
The maximum possible Perron value of a tournament matrix T of order n is (n−1)/2, attained when the row sums of T are all equal [1]. 0 0 ⋅ ⋅ ⋅ 0 0 n×n is the tournament matrix of order 2n which maximizes the Perron value. This conjecture has recently been confirmed in [3]. Let ρ2n = ρ(B2n) denote the Perron value of the Brualdi-Li matrix of order 2n. Kirkland [6] conjectured that 2n(n − 1/2 − ρ2n) is monotonically decreasing for all n. By Theorem 1 and Lemma 4, one can prove the following corollaries.
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