Hermitian $A_\alpha$-矩阵的局部与全局谱界
Local and Global Spectral Bounds for Hermitian $A_α$-Matrices
- Dr B R Ambedkar National Institute of Technology Jalandhar（贾朗达尔 B.R.安贝德卡尔国立理工学院）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文为混合图的Hermitian $A_\alpha$-矩阵建立局部与全局谱界，利用谱矩和局部压缩方法给出特征值上下界及谱宽度估计，并应用于图滤波收敛和残差GNN稳定性。
AI中文摘要：
我们为混合图的Hermitian $A_\alpha$-矩阵建立了局部与全局谱界。利用前三个谱矩，我们得到了最大特征值的上界，该上界是某个显式三次多项式的最大实零点。极端特征值的逐顶点估计为谱宽度提供了新的下界，其中包括一个严格改进普通图现有基于度估计的界。一种局部二维压缩方法产生了涉及邻居度数据和带权三角形（gain-weighted triangles）的谱宽度界。该界对星的每个混合定向都是精确的，并且独立于已知的Zagreb指标界。我们还推导了两个关于最小特征值之和的互补上界。作为进一步的结果，这些谱估计为Richardson图滤波提供了可计算的收敛保证，并为有向网络上的残差图神经网络层提供了稳定性证书。数值示例说明了所提出界的尖锐性和相互不可比性。
英文摘要：
We establish local and global spectral bounds for Hermitian $A_α$-matrices of mixed graphs. Using the first three spectral moments, we obtain an upper bound for the largest eigenvalue as the largest real zero of an explicit cubic polynomial. Vertexwise estimates for the extreme eigenvalues yield new lower bounds for the spectral spread, including a bound that strictly improves an existing degree-based estimate for ordinary graphs. A local two-dimensional compression produces a spread bound involving neighbour-degree data and gain-weighted triangles. This bound is exact for every mixed orientation of a star and is independent of a known Zagreb-index bound. We also derive two complementary upper bounds for sums of the smallest eigenvalues. As further consequences, the spectral estimates provide a computable convergence guarantee for Richardson graph filtering and a stability certificate for residual graph-neural-network layers on directed networks. Numerical examples illustrate the sharpness and mutual incomparability of the proposed bounds.
