曲率、图乘积与Ricci平坦性
Curvatures, graph products and Ricci flatness
AI总结：
本文比较了组合图上的Ollivier Ricci曲率和Bakry-Émery曲率，证明了在特定条件下前者非负性蕴含后者，并研究了Ricci平坦性及曲率非负性在图乘积下的保持情况。
摘要：
在本文中，我们比较了组合图上的Ollivier Ricci曲率和Bakry-Émery曲率概念，并讨论了它们与各种类型Ricci平坦性的联系。我们证明了在无三角形条件和附加的入度条件下，Ollivier Ricci曲率的非负性意味着Bakry-Émery曲率的非负性。我们还提供了例子说明该结果的两个条件都是必要的。我们研究了与图乘积的关系，并证明Ricci平坦性在所有自然乘积下都得到保持。虽然这两种曲率的非负性在笛卡尔乘积下都得到保持，但我们证明在强乘积的情况下，Ollivier Ricci曲率的非负性仅对水平边和垂直边得到保持。我们还证明了所有周长为$4$的距离正则图都达到其最大可能的曲率值。
英文摘要：
In this paper, we compare Ollivier Ricci curvature and Bakry-Émery curvature notions on combinatorial graphs and discuss connections to various types of Ricci flatness. We show that non-negativity of Ollivier Ricci curvature implies non-negativity of Bakry-Émery curvature under triangle-freeness and an additional in-degree condition. We also provide examples that both conditions of this result are necessary. We investigate relations to graph products and show that Ricci flatness is preserved under all natural products. While non-negativity of both curvatures are preserved under Cartesian products, we show that in the case of strong products, non-negativity of Ollivier Ricci curvature is only preserved for horizontal and vertical edges. We also prove that all distance-regular graphs of girth $4$ attain their maximal possible curvature values.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
