Eigenspaces of Tournament Matrices

NEW

Problem: Reconstructed statement: for an even composite $N=nm$, let $\eta$ be the smallest prime factor of $N$. If $T\in\mathcal T_N$ is not permutation-similar to the Brualdi-Li matrix $B_N$, and if there is a Brualdi-Li tally tournament $\mathcal T=T(A,B_{N/\eta})$, with $A$ regular, then

$$\rho(T)\le \rho(\mathcal T)<\rho(B_N).$$

Here $B_{2r}=\begin{pmatrix}L&L^t\\ L^t+I_r&L\end{pmatrix}$, with $L$ the strictly lower triangular $r\times r$ tournament matrix, and $T(A,B)=B\otimes J+I\otimes A$.

Result: The literal statement is vacuous. Since $N$ is even, $\eta=2$. A Brualdi-Li tally of order $N/2$ would require a regular tournament $A\in\mathcal T_2$, but no such tournament exists: its row sums would have to equal $(2-1)/2=1/2$.

The natural repair is to take $\eta$ to be the smallest odd prime divisor of $N$. That repaired conjecture is false.

Counterexample: take $N=12$, $\eta=3$. Let $\mathcal T=T(C_3,B_4)$, where $C_3$ is the directed 3-cycle. Since $C_3$ is regular,

$$\rho(\mathcal T)=3\rho(B_4)+1.$$

Now $\rho(B_4)$ is the positive root of $x^4-2x-1$, so $\rho(\mathcal T)$ is the positive root of

$$q(x)=x^4-4x^3+6x^2-58x-26.$$

Since $q(5)<0$, $q(26/5)>0$, and $q'(x)>0$ for $x\ge5$, we have

$$\rho(\mathcal T)<26/5=5.2.$$

Let $T$ be obtained from $B_{12}$ by reversing the edge between vertices $2$ and $7$. Then $T\in\mathcal T_{12}$, and a direct determinant computation gives

$$\chi_T(x)=x^{12}-70x^9-307x^8-796x^7-1358x^6-1644x^5-1429x^4-876x^3-360x^2-89x-10.$$

Also,

$$\chi_T(109/20)<0,$$

while $\chi_T(x)\to+\infty$, so $T$ has a positive eigenvalue $>109/20=5.45$. Hence

$$\rho(T)>5.45>5.2>\rho(\mathcal T).$$

Finally, $T$ is not permutation-similar to $B_{12}$, since their characteristic polynomials differ. Thus the repaired conjecture is disproved.

Citation: Statement reconstructed from James L. Burk Jr., Eigenspaces of Tournament Matrices, Section 8.1. The counterexample above is direct. Context: S. W. Drury, “Solution of the conjecture of Brualdi and Li,” Linear Algebra Appl. 436 (2012), 3392–3399.