图的体积增长、曲率和Buser型不等式
Volume growth, curvature, and Buser-type inequalities in graphs
摘要：
研究了离散空间中度量球的体积增长与半径的关系，重点讨论了体积增长与离散曲率的关系。我们在所谓的奥利维尔曲率的下界下改进了体积增长边界，并在其他类型的离散里奇曲率下讨论了类似的结果。继最近的工作在连续设置的黎曼流形（由第一作者），然后，我们绑定的特征值的拉普拉斯算子的一个图形的范围内的体积增长。特别地，图的谱隙可以使用加权离散Hardy不等式来界定，并且图的高特征值可以由三对角矩阵的特征值乘以乘法因子来界定，这两者都仅取决于图的体积增长。作为一个直接的应用，我们将本征值与Cheeger等周常数联系起来。利用这些方法，我们描述了一类图的Cheeger不等式是紧的第二特征值。我们还描述了一种方法证明Buser不等式的图形，特别是在一个下界假设曲率。
英文摘要：
We study the volume growth of metric balls as a function of the radius in discrete spaces, and focus on the relationship between volume growth and discrete curvature. We improve volume growth bounds under a lower bound on the so-called Ollivier curvature, and discuss similar results under other types of discrete Ricci curvature. Following recent work in the continuous setting of Riemannian manifolds (by the first author), we then bound the eigenvalues of the Laplacian of a graph under bounds on the volume growth. In particular, the spectral gap of the graph can be bounded using a weighted discrete Hardy inequality and the higher eigenvalues of the graph can be bounded by the eigenvalues of a tridiagonal matrix times a multiplicative factor, both of which only depend on the volume growth of the graph. As a direct application, we relate the eigenvalues to the Cheeger isoperimetric constant. Using these methods, we describe classes of graphs for which the Cheeger inequality is tight on the second eigenvalue. We also describe a method for proving Buser's inequality in graphs, particularly under a lower bound assumption on curvature.
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