Kemeny常数：通过矩阵压缩与特征值交错

中文摘要 AI 辅助

Kemeny常数量化了随机游走到达随机选择顶点所需的期望时间，捕捉了马尔可夫链的全局性质。我们发展了一个矩阵分析框架，利用归一化邻接矩阵的度加权压缩、收缩不等式和特征值交错来界定有限连通加权图的Kemeny常数。我们的主要划分定理给出了基于压缩谱的下界，并具有完整的等式刻画，将结构图信息转化为谱可计算的估计。例如，当应用于适当的颜色划分时，所提出的方法产生了关于色数的Kemeny常数的尖锐下界 \\[ K(G)\ge n-2+\frac{1}{\chi(G)}, \\] 这扩展了Ciardo、Dahl和Kirkland（2022）的二部图界到任意色数，并且与Chung（1997）的归一化Hoffman界不可比较。对于无权图，我们还刻画了该界的所有等式情形。我们进一步通过推导涉及归一化割和电导的NP难图问题的谱界，展示了所提出的矩阵框架的威力。我们的结果包括渐近尖锐的电导界、来自主子矩阵和商矩阵的双侧交错界，以及在保持连通性的多边删除下Kemeny常数变化的估计。

英文摘要

Kemeny's constant quantifies the expected time for a random walk to reach a randomly chosen vertex, capturing global properties of a Markov chain. We develop a matrix-analytic framework for bounding Kemeny's constant of a finite connected weighted graph using degree-weighted compressions of the normalized adjacency matrix, pinching inequalities, and eigenvalue interlacing. Our main partition theorem gives lower bounds in terms of the compressed spectrum, with complete equality characterizations, and converts structural graph information into spectrally computable estimates. For instance, when applied to proper color partitions, the proposed method yields a sharp lower bound on Kemeny's constant in terms of the chromatic number \[ K(G)\ge n-2+\frac{1}{χ(G)}, \] which extends a bipartite bound of Ciardo, Dahl, and Kirkland (2022) to arbitrary chromatic number, and which is incomparable with the normalized Hoffman bound by Chung (1997). For unweighted graphs, we also characterize all equality cases of this bound. We further illustrate the power of the proposed matrix framework by deriving spectral bounds for NP-hard graph problems involving normalized cuts and conductance. Our results include an asymptotically sharp conductance bound, two-sided interlacing bounds from principal submatrices and quotient matrices, and estimates for the change in Kemeny's constant under connectivity-preserving deletion of multiple edges.

发表机构

• Eindhoven University of Technology（埃因霍温理工大学）
• Vrije Universiteit Brussel（布鲁塞尔自由大学）
• Universitat Politècnica de Catalunya（加泰罗尼亚理工大学）
• Hamburg University of Technology（汉堡工业大学）
• Universidad de Valladolid（巴利亚多利德大学）

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