Discrete Curvature and Abelian Groups
Discrete Curvature and Abelian Groups
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DOI：
10.4153/cjm-2015-046-8
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发表时间：
2015-01
期刊：
影响因子：
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通讯作者：
B. Klartag;G. Kozma;P. Ralli;P. Tetali
中科院分区：
文献类型：
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作者：
B. Klartag;G. Kozma;P. Ralli;P. Tetali
摘要研究了图上的一个自然离散Bochner型不等式，并探讨了它作为离散空间中“曲率”概念的价值。这个离散版本的所谓的${{\Gamma }_{2}}$ -演算（Bakry-Émery）的一个吸引人的特点似乎是，它是相当简单的计算这个概念的曲率参数的几个特定的图的兴趣，特别是，阿贝尔群，切片的超立方体，和对称群下的各种集的发电机。我们进一步发展这一概念，通过推导Buser型不等式（à la Ledoux），与图相关的函数和等周常数。我们的推导提供了Cheeger常数的严格界限（即，边等周常数）的谱隙，特别是，一类交换Cayley图的非负曲率，一个结果的独立利益。
Abstract We study a natural discrete Bochner-type inequality on graphs, and explore its merit as a notion of “curvature” in discrete spaces. An appealing feature of this discrete version of the so-called ${{\Gamma }_{2}}$ -calculus (of Bakry-Émery) seems to be that it is fairly straightforward to compute this notion of curvature parameter for several specific graphs of interest, particularly, abelian groups, slices of the hypercube, and the symmetric group under various sets of generators. We further develop this notion by deriving Buser-type inequalities (à la Ledoux), relating functional and isoperimetric constants associated with a graph. Our derivations provide a tight bound on the Cheeger constant (i.e., the edge-isoperimetric constant) in terms of the spectral gap, for graphs with nonnegative curvature, particularly, the class of abelian Cayley graphs, a result of independent interest.