On Some Properties for the Sequence of Brualdi-Li Matrices

Abstract

Let 𝐵_2𝑛 denote the Brualdi-Li matrix, and let 𝜌_2𝑛=𝜌(𝐵_2𝑛) denote the Perron value of the Brualdi-Li matrix of order 2𝑛. We prove that 2𝑛(𝑛−1/2−𝜌_2𝑛) is monotonically decreasing for all 𝑛 and 𝜌_2𝑛<𝑛−1/2−(𝑒²−1)/4(𝑒²+1)𝑛, where 𝑒=2.718281828459045….

1. Introduction

A tournament matrix of order 𝑛 is a (0,1) matrix 𝑇 satisfying the equation 𝑇+𝑇^𝑡=𝐽−𝐼, where 𝐽 is the all ones matrix, 𝐼 is the identity matrix, and 𝑇^𝑡 is the transpose of 𝑇. If 𝑛 is odd, then 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]. However, if 𝑛 is even, it is difficult to attain. Brualdi and Li [2] conjectured that the matrix ℬ_2𝑛=𝑈_𝑛𝑈^𝑡_𝑛𝐼+𝑈^𝑡_𝑛𝑈_𝑛, (1) where 𝑈_𝑛 is strictly upper triangular tournament matrix (all of its entries above the main diagonal are equal to one), that is,𝑈_𝑛=011⋯1001⋯1⋮⋱⋱⋱⋮00⋯0100⋯00_𝑛×𝑛, (2) is the tournament matrix of order 2𝑛 which maximizes the Perron value. This conjecture has recently been confirmed in [3].

The matrix ℬ_2𝑛 has been dubbed by the Brualdi-Li matrix [4]. Let 𝜌_2𝑛=𝜌(𝐵_2𝑛) denote the Perron value of the Brualdi-Li matrix of order 2𝑛.

It is well known that 𝑛−1<𝜌_2𝑛<𝑛.

Friedland [4] has paid particular attention to the behavior of the Perron value of ℬ_2𝑛. Friedland and Katz [5] posed the problem of determining the behavior of the sequence 2𝑛(𝑛−1/2−𝜌_2𝑛). Kirkland [6] conjectured that 2𝑛(𝑛−1/2−𝜌_2𝑛) is monotonically decreasing for all 𝑛. We investigate this conjecture and obtain the following results.

Theorem 1. Let 𝑛≥2 be an integer, and let 𝜌_2𝑛=𝜌(ℬ_2𝑛) be the Perron value of the Brualdi-Li matrix ℬ_2𝑛. Then 2𝑛(𝑛−1/2−𝜌_2𝑛) is monotonically decreasing for all 𝑛.

By Theorem 1 and Lemma 4, one can easily prove the following corollaries.

Corollary 2. Let 𝑛≥2 be an integer, and let 𝜌_2𝑛=𝜌(ℬ_2𝑛) be the Perron value of the Brualdi-Li matrix ℬ_2𝑛. Then 𝜌_2𝑛1<𝑛−2−𝑒²−14𝑒²𝑛,+1 (3) where 𝑒=2.718281828459045….

In [6], Kirkland proved that lim_𝑛→+∞𝑛³1𝑛−2−𝛾4𝑛−𝜌_2𝑛=𝛾²+𝑒16²3𝛾+112𝑒²+1², (4) where 𝛾=(𝑒²−1)/(𝑒²+1).

We make the following conjecture and remark that it is supported by computations for 𝑛≤100.

Conjecture 3. 𝑛³(𝑛−1/2−𝛾/4𝑛−𝜌_2𝑛) is monotonically decreasing for all 𝑛>1.

We give some lemmas in Section 2 and prove Theorem 1 in Section 3.

2. Some Lemmas

2.1. Intermediate Value Theorem

If 𝑓 is a real-valued continuous function on the closed interval [𝑎,𝑏], and 𝜇 is a number between 𝑓(𝑎) and 𝑓(𝑏), then there is 𝜉∈(𝑎,𝑏) such that 𝑓(𝜉)=𝜇. where (𝑎,𝑏) is open interval.

For two variables, the Implicit Function Theorem is as follows.

2.2. Implicit Function Theorem

If a real-valued function 𝐹 is defined on an open disk containing point (𝑥₀,𝑦₀), satisfying(i)𝐹(𝑥₀,𝑦₀)=0,(ii)𝜕𝐹/𝜕𝑦|_{(𝑥0,𝑦0)}≠0,(iii)𝜕𝐹/𝜕𝑥 and 𝜕𝐹/𝜕𝑦 are continuous on the open disk,then the equation 𝐹(𝑥,𝑦)=0 defines 𝑦 as a function of 𝑥 near the point (𝑥₀,𝑦₀) and the derivative of this function is given by 𝑑𝑦𝑑𝑥=−𝜕𝐹/𝜕𝑥,𝜕𝐹/𝜕𝑦 (5) where 𝜕𝐹/𝜕𝑥 and 𝜕𝐹/𝜕𝑦 indicate the partial derivatives of 𝐹 with respect to 𝑥 and 𝑦.

Lemma 1 (see [6]). Let 𝑛≥2 be an integer, and let 𝜌_2𝑛=𝜌(ℬ_2𝑛) be the Perron value of the Brualdi-Li matrix ℬ_2𝑛. Then 2𝜌²_2𝑛+2(1−𝑛)𝜌_2𝑛+1−𝑛1+𝜌⁻¹_2𝑛^2𝑛+1−1=0. (6)

Let 𝐹=𝐹(𝑚,𝜌)=(2𝜌²+2(1−𝑚)𝜌+1−𝑚)((1+𝜌⁻¹)^2𝑚+1)−1 and 𝐷={(𝑚,𝜌)∣𝑚>2, 𝑚−1<𝜌<𝑚, 𝑚 and 𝜌 are real numbers}, then the partial derivatives of 𝐹 are 𝜕𝐹𝜕𝑚=−2𝜌+11+𝜌^−12𝑚+1+22𝜌²()×+21−𝑚𝜌+1−𝑚1+𝜌^−12𝑚ln1+𝜌⁻¹,𝜕𝐹𝜕𝜌=22𝜌−𝑚+11+𝜌^−12𝑚+1−2𝑚𝜌⁻²2𝜌²()×+21−𝑚𝜌+1−𝑚1+𝜌^{−12𝑚−1}. (7) Evidently, 𝜕𝐹/𝜕𝑚 and 𝜕𝐹/𝜕𝜌 are derivable on the open disk 𝐷, and 𝜕𝐹/𝜕𝜌=2(2𝜌−𝑚+1)+2((2𝜌−𝑚+1)(1+𝜌⁻¹) −𝑚𝜌⁻²(2𝜌²+2(1−𝑚)𝜌+1−𝑚))⋅(1+𝜌⁻¹)^2𝑚−1>0, (𝑚,𝜌)∈𝐷. By the Implicit Function Theorem, the equation 𝐹(𝑚,𝜌)=0 defines 𝜌 as a function in terms of 𝑚 and the derivative of this function is given by 𝑑𝜌𝑑𝑚=−𝜕𝐹/𝜕𝑚.𝜕𝐹/𝜕𝜌 (8)

Lemma 2. Let 𝑚≥3 be a real number, and let 𝜌 be a function in terms of 𝑚 satisfying the equation 𝐹(𝑚,𝜌)=0. Then 1𝑚−2−115𝑚<𝜌<𝑚−2−10,53𝑚 (9) where 𝐹(𝑚,𝜌)=(2𝜌²+2(1−𝑚)𝜌+1−𝑚)((1+𝜌⁻¹)^2𝑚+1)−1, (𝑚,𝜌)∈𝐷.

Proof. For fixed real number 𝑚≥3, 𝐹(𝑚,𝜌) is continuous real function in terms of 𝜌.
Let 𝜌=𝑚−1/2−1/5𝑚 and 𝜌=𝑚−1/2−10/53𝑚. It is easy to show that the real functions 𝑦₁(𝑥)=(1+(1/(𝑥−1/2−1/5𝑥)))^2𝑥 and 𝑦₂(𝑥)=(1+(1/(𝑥−1/2−10/53𝑥)))^2𝑥 are monotonically decreasing for real number 𝑥>2. Thus 𝐹𝑚,𝜌=2𝜌²+2(1−𝑚)𝜌×+1−𝑚1+𝜌^−12𝑚=21+1−1𝑚−2−15𝑚²()1+21−𝑚𝑚−2−1⋅15𝑚+1−𝑚1+𝑚−1/2−1/5𝑚^2𝑚=1+1−12−25+25²𝑚²×11+𝑚−1/2−1/5𝑚^2𝑚≤1+1−12−25+25²⋅3²×11+3−1/2−1/15⁶+1−1=−0.0319⋯<0. (10)
Notice that lim_𝑥→+∞11+𝑥−1/2−10/53𝑥^2𝑥=𝑒²=7.38905609893065…, (11) where 𝑒=2.718281828459045…. We have 𝐹𝑚,𝜌=21𝑚−2−1053𝑚²()×1+21−𝑚𝑚−2−10⋅153𝑚+1−𝑚1+𝑚−1/2−10/53𝑚^2𝑚=1+1−12−20+53200(53𝑚)²×11+𝑚−1/2−10/53𝑚^2𝑚>1+1−12−20𝑒53²+1−1>0. (12)
Hence 𝐹(𝑚,𝜌)𝐹(𝑚,𝜌)<0. By the Intermediate Value Theorem, there is 𝜌∈(𝜌,𝜌) such that 𝐹(𝑚,𝜌)=0; that is, 1𝑚−2−115𝑚<𝜌<𝑚−2−10.53𝑚 (13) We are done.

Lemma 3. Let 𝑚≥3 be a real number, and let 𝜌 be a function in terms of 𝑚 satisfying the equation 𝐹(𝑚,𝜌)=0, and let 𝐺=𝐺(𝑚,𝜌)=2𝜌²+2(1−𝑚)𝜌+1−𝑚, (𝑚,𝜌)∈𝐷. Then 1+21025𝑚²93−1062002089𝑚²<9<𝐺𝑚,𝜌1−𝐺𝑚,𝜌−21025𝑚²13+1062002089𝑚². (14)

Proof. For fixed real number 𝑚≥3, 𝐺(𝑚,𝜌) is continuous real function in terms of 𝜌. Let 𝜌=𝑚−1/2−1/5𝑚 and 𝜌=𝑚−1/2−10/53𝑚; then 𝐺𝑚,𝜌=1+21025𝑚²,𝐺𝑚,𝜌=13+1062002089𝑚². (15) By Lemma 2, we have 𝐺𝑚,𝜌<𝐺𝑚,𝜌=2𝜌²+2(1−𝑚)𝜌+1−𝑚<𝐺𝑚,𝜌. (16) Hence 1+21025𝑚²<𝐺𝑚,𝜌<10+1062002089𝑚²; (17) that is, 0<93−1062002089𝑚²<9<1−𝐺𝑚,𝜌−21025𝑚². (18) We have the desired result.

Lemma 4 (see [6]). There is an 𝑀 such that the sequence 2𝑛(𝑛−1/2−𝜌_2𝑛) decreases monotonically to its limit (𝑒²−1)/2(𝑒²+1) for 𝑛>𝑀.

3. Proof of Theorem 1

In Section 2, 𝐹=𝐹(𝑚,𝜌)=(2𝜌²+2(1−𝑚)𝜌+1−𝑚)((1+𝜌⁻¹)^2𝑚+1)−1 and 𝐺=𝐺(𝑚,𝜌)=2𝜌² +2(1−𝑚)𝜌+1−𝑚, (𝑚,𝜌)∈𝐷, and 𝜌 is the function in terms of 𝑚 satisfying the equation 𝐹(𝑚,𝜌)=0. Clearly, 𝐹 and 𝐺 are the functions in terms of 𝑚.

Let 𝑚≥3 be real number. Since 𝐹(𝑚,𝜌)=0, we have 1+𝜌^−12𝑚=1−𝐺𝑚,𝜌𝐺,𝑚,𝜌𝑑𝐹𝑑𝑚=0. (19) Further 𝑑𝐹=𝑑𝑑𝑚𝐺𝑑𝑚1+𝜌^−12𝑚𝑑+1=𝐺𝑑𝑚1+𝜌^−12𝑚+1+1+𝜌^−12𝑚+1𝑑𝐺𝑑𝑚=2𝐺1+𝜌^−12𝑚ln1+𝜌⁻¹−𝑚𝜌1+𝜌𝑑𝜌+𝑑𝑚1+𝜌^−12𝑚+14𝜌+2−2𝑚𝑑𝜌()𝑑𝑚−2𝜌−1=21−𝐺ln1+𝜌⁻¹−𝑚𝜌1+𝜌𝑑𝜌+1𝑑𝑚𝐺4𝜌+2−2𝑚𝑑𝜌=1𝑑𝑚−2𝜌−1𝐺2(1−𝐺)𝐺ln1+𝜌⁻¹()𝐺𝑚−21−𝐺𝜌1+𝜌𝑑𝜌+𝑑𝑚4𝜌+2−2𝑚𝑑𝜌=1𝑑𝑚−2𝜌−1𝜌𝐺×()1+𝜌2𝜌1+𝜌1−𝐺𝐺ln1+𝜌⁻¹−2𝑚(1−𝐺)𝐺𝑑𝜌𝑑𝑚+𝜌1+𝜌4𝜌+2−2𝑚𝑑𝜌=1𝑑𝑚−2𝜌−1𝜌𝐺×)1+𝜌2𝜌1+𝜌(1−𝐺𝐺ln1+𝜌⁻¹+2𝑚(𝐺−1)𝐺𝑑𝜌𝑑𝑚+𝜌1+𝜌4𝜌+2−2𝑚𝑑𝜌𝑑𝑚−𝜌1+𝜌2𝜌+1=0. (20) Thus 2𝑚(𝐺−1)𝐺+𝜌1+𝜌4𝜌+2−2𝑚𝑑𝜌()𝑑𝑚=2𝜌1+𝜌𝐺−1𝐺ln1+𝜌⁻¹+𝜌1+𝜌2𝜌+1. (21) We have 𝑑𝜌=()𝑑𝑚2𝜌1+𝜌𝐺−1𝐺ln1+𝜌⁻¹×()+𝜌1+𝜌2𝜌+12𝑚𝐺−1𝐺+𝜌1+𝜌4𝜌+2−2𝑚⁻¹. (22) Hence 𝑑𝜌𝑑𝑚−2+2𝜌+1=()2𝑚2𝜌1+𝜌𝐺−1𝐺ln1+𝜌⁻¹+𝜌1+𝜌2𝜌+1()+2𝑚𝐺−1𝐺+𝜌1+𝜌4𝜌+2−2𝑚2𝜌+1−4𝑚=()2𝑚2𝑚2𝜌1+𝜌𝐺−1𝐺ln1+𝜌⁻¹×𝜌()𝐺+𝜌1+𝜌2𝜌+12𝑚1+𝜌4𝜌+2−2𝑚−2𝑚1−𝐺⁻¹+()××𝜌()𝐺2𝑚𝐺−1𝐺+𝜌1+𝜌4𝜌+2−2𝑚2𝜌+1−4𝑚2𝑚1+𝜌4𝜌+2−2𝑚−2𝑚1−𝐺⁻¹. (23) Let 𝐴=𝜌(1+𝜌)(4𝜌+2−2𝑚)−2𝑚(1−𝐺)𝐺 and 𝐵=2𝑚2𝜌1+𝜌(𝐺−1)𝐺ln1+𝜌⁻¹+()×;+𝜌1+𝜌2𝜌+12𝑚𝐺−1𝐺+𝜌1+𝜌4𝜌+2−2𝑚2𝜌+1−4𝑚 (24) then 𝑑𝜌𝑑𝑚−2+2𝜌+1=𝐵2𝑚,()2𝑚𝐴𝐵=4𝑚𝜌1+𝜌𝐺−1𝐺ln1+𝜌⁻¹()𝐺()+2𝑚𝜌1+𝜌2𝜌+1+2𝑚2𝜌+1−4𝑚𝐺−1+𝜌1+𝜌4𝜌+2−2𝑚2𝜌+1−4𝑚=2𝑚𝜌1+𝜌2𝜌+1+𝜌1+𝜌4𝜌+2−2𝑚2𝜌+1−4𝑚+4𝑚𝜌1+𝜌𝐺−1𝐺ln1+𝜌⁻¹)𝐺𝑚+()×+2𝑚2𝜌+1−4𝑚(𝐺−1=2𝜌1+𝜌2𝜌+12𝜌+1−𝑚2𝜌+1−4𝑚+2𝑚𝐺1−𝐺4𝑚−2𝜌−1−2𝜌1+𝜌ln1+𝜌⁻¹. (25) Let 𝐶=𝑚(2𝜌+1)+(2𝜌+1−𝑚)(2𝜌+1−4𝑚) and 𝐷=4𝑚−2𝜌−1−2𝜌(1+𝜌)ln(1+𝜌⁻¹), then 𝐵=2𝜌1+𝜌𝐶+2𝑚𝐺(1−𝐺)𝐷. (26) Obviously, +=𝐶=𝑚2𝜌+12𝜌+1−𝑚2𝜌+1−4𝑚2𝜌+1−2𝑚²>0. (27) By Lemma 2 and Taylor’s Theorem, we have 𝐷=4𝑚−2𝜌−1−2𝜌1+𝜌ln1+𝜌⁻¹1>4𝑚−2𝜌−1−2𝜌1+𝜌𝜌−12𝜌²+13𝜌³1=4𝑚−4𝜌−2+−23𝜌3𝜌²1>4𝑚−4𝑚−2−10153𝑚−2+−23𝜌3𝜌²=123𝜌1−𝜌+40.53𝑚 (28) Due to 𝜌>𝑚−1≥2, then 𝐷>0. (29) By Lemmas 2 and 3, ()𝐺>()𝑚(4())9𝐴=𝜌1+𝜌4𝜌+2−2𝑚−2𝑚1−𝐺𝑚−1𝑚−1+2−2𝑚−2𝑚−21025𝑚²13+1062002089𝑚²=2𝑚(𝑚−1)²−9−21025𝑚²13+1062002089𝑚²>0. (30) By Lemma 3 and (25)–(30), we have 𝐵/2𝑚𝐴>0 for 𝑚≥3. Hence (𝑑/𝑑𝑚)(2𝑚(𝑚−1/2−𝜌))<0, where 𝑚≥3 is a real number.

To sum up, we obtain the following result.

Let 𝑚≥3 be a real number, and let 𝜌 be a function of 𝑚 satisfying the equation 𝐹(𝑚,𝜌)=0, (𝑚,𝜌)∈𝐷; then 2𝑚(𝑚−1/2−𝜌) is monotonically decreasing function for all 𝑚.

Note that 𝜌_2𝑛=𝜌(ℬ_2𝑛) is the Perron value of the Brualdi-Li matrix ℬ_2𝑛, by Lemma 1, 𝜌_2𝑛 satisfy the equation 𝐹𝑛,𝜌_2𝑛=2𝜌²_2𝑛+2(1−𝑛)𝜌_2𝑛×+1−𝑛1+𝜌⁻¹_2𝑛^2𝑛+1−1=0, (31) where 𝑛≥3 is an integer. It is easy to see that 2𝑛(𝑛−1/2−𝜌_2𝑛) is monotonically decreasing for all 𝑛≥3.

Since 12×22−2−𝜌_2×2()1=41.5−1.3953>2×33−2−𝜌_2×3=6(2.5−2.4340), (32) evidently, it follows that 2𝑛(𝑛−1/2−𝜌_2𝑛) is monotonically decreasing for all integers 𝑛≥2. We complete the proof of Kirkland’s conjecture.

Acknowledgment

This work is supported by the Natural Science Foundation of Guangdong Province, China (no. S2012010009759).