图的粗里奇曲率的精确和渐近结果
Exact and Asymptotic Results on Coarse Ricci Curvature of Graphs
摘要：
里奇曲率是由Ollivier在度量测度空间的一般框架下提出的，近年来在图的背景下得到了广泛的研究。本文证明了二部图和围长至少为5的图的Ollivier里奇曲率的上界。我们还证明了一个一般的下界的里奇曲率的大小方面的最大匹配在一个适当的子图。作为结果，我们刻画了围长为5的Ricci平坦图。利用我们的一般下界和Birkhoff-von Neumann定理，给出了围长为4的Ricci平坦正则图的结构的一个充要条件.最后，我们得到了随机二部图$G(n,n, p)$和随机图$G(n, p)$在$p$的不同区域下的渐近里奇曲率.
英文摘要：
Ricci curvature was proposed by Ollivier in a general framework of metric measure spaces, and it has been studied extensively in the context of graphs in recent years. In this paper we prove upper bounds for Ollivier's Ricci curvature for bipartite graphs and for the graphs with girth at least 5. We also prove a general lower bound on the Ricci curvature in terms of the size of the maximum matching in an appropriate subgraph. As a consequence, we characterize the Ricci-flat graphs of girth 5. Moreover, using our general lower bound and the Birkhoff-von Neumann theorem, we give a necessary and sufficient condition for the structure of Ricci-flat regular graphs of girth 4. Finally, we obtain the asymptotic Ricci curvature of random bipartite graphs $G(n,n, p)$ and random graphs $G(n, p)$, in various regimes of $p$.
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