Bilu--Linial 猜想的改进界：通过混合行列式多项式的谱恢复
Improved Bounds for the Bilu--Linial Conjecture via Spectral Recovery from Mixed Determinantal Polynomials
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- School of Mathematics and Statistics, Fuzhou University（福州大学数学与统计学院）
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中文总结 AI 辅助
本文通过构造正矩阵值测度并利用短游走估计，将 Bilu--Linial 猜想中满足混合根条件的符号的谱半径界改进为 $(3+\sqrt5)/2\sqrt{d-1}$，并推广到多矩阵情形。
中文摘要 AI 辅助
Bilu--Linial 猜想询问：每个 $d\geq 2$ 的有限 $d$-正则图是否都存在一个边符号 $\sigma$，使得其符号邻接矩阵 $A_\sigma$ 的谱半径至多为 $2\sqrt{d-1}$。我们证明，任何满足混合根条件 $r_{A_\sigma}\leq\sqrt{2(d-1)}$ 的符号都满足 \\[ \rho(A_\sigma) < \frac{3+\sqrt5}{2}\sqrt{d-1}, \\] 其中 $r_{A_\sigma}$ 是混合行列式多项式 $\chi[A_\sigma,-A_\sigma]$ 的最大根。Ravichandran 和 Srivastava 的交错定理保证了满足混合根条件的符号存在，因此我们的结果改进了他们双边谱界中的系数 $2\sqrt2$。在证明中，我们构造了一个正矩阵值概率测度，其支撑在 $\chi[A_{\sigma},-A_{\sigma}]$ 的根上。二阶矩给出了一个简单的矩阵不等式 $A_{\sigma}^2 + dI \preceq 4r_{A_\sigma}^2I$，由此得到初步系数 $\sqrt{7}$。四阶矩的估计利用短游走的信息得到系数 $(3+\sqrt{5})/2$。在更多图结构假设下，对于无三角形图，系数改进为 $\sqrt6$；对于围长至少为 5 的图，系数改进为 $\sqrt{(5+3\sqrt5)/2}$。作为一个独立有趣的结果，我们将构造推广到零对角 Hermitian 矩阵 $A_1,\ldots,A_k$ 的 $\chi[A_1,\ldots,A_k]$，并显式计算了前两阶矩。最后，$K_8$ 的一个显式符号表明，仅混合根条件无法保证系数低于 $(4+\sqrt5)/\sqrt6$。
英文摘要
The Bilu--Linial conjecture asks whether every finite $d$-regular graph with $d \geq 2$ admits an edge signing $σ$ whose signed adjacency matrix $A_σ$ has spectral radius at most $2\sqrt{d-1}$. We prove that every signing meeting the mixed-root condition $r_{A_σ}\leq\sqrt{2(d-1)}$ satisfies \[ ρ(A_σ) < \frac{3+\sqrt5}{2}\sqrt{d-1}, \] where $r_{A_σ}$ is the largest root of the mixed determinantal polynomial $χ[A_σ,-A_σ]$. The interlacing theorem of Ravichandran and Srivastava guarantees a signing satisfying the mixed-root condition, so our result improves the coefficient $2\sqrt2$ in their two-sided spectral bound. In the proof, we construct a positive matrix-valued probability measure supported on the roots of $χ[A_σ,-A_σ]$. The second moment gives a simple matrix inequality $A_σ^2 + dI \preceq 4r_{A_σ}^2I$, which yields a preliminary coefficient $\sqrt{7}$. Estimates for the fourth moment use information about short walks to obtain the coefficient $(3+\sqrt{5})/2$. With more graph structural assumptions, the coefficient improves to $\sqrt6$ for triangle-free graphs and to $\sqrt{(5+3\sqrt5)/2}$ for graphs of girth at least five. As a result of independent interest, we extend the construction to $χ[A_1,\ldots,A_k]$ for Hermitian matrices $A_1,\ldots,A_k$ with zero diagonal, and compute the first two moments explicitly. Finally, an explicit signing of $K_8$ shows that the mixed-root condition alone cannot guarantee a coefficient below $(4+\sqrt5)/\sqrt6$.
