In this paper we investigate some properties of the Fiedler vector, the so-called first non-trivial eigenvector of the Laplacian matrix of a graph. There are important results about the Fiedler vector to identify spectral cuts in graphs but far less is known about its extreme values and points. We propose a few results and conjectures in this direction. We also bring two concrete contributions, i) by defining a new measure for graphs that can be interpreted in terms of extremality (inverse of centrality), ii) by applying a small perturbation to the Fiedler vector of cerebral shapes such as the corpus callosum to robustify their parameterization.

翻译：本文研究了图拉普拉斯矩阵的第一非平凡特征向量——菲德勒向量的若干性质。关于利用菲德勒向量识别图的谱割已有重要成果，但其极值与极值点的特性尚不明确。我们在此方向上提出若干结论与猜想，并贡献两项具体应用：i) 定义一种新的图测度，可解释为极端性（中心性的逆度量）；ii) 对脑部形态（如胼胝体）的菲德勒向量施加小扰动，以增强其参数化的鲁棒性。