A version of Bakry-Émery Ricci flow on a finite graph

Abstract

In this paper, we study the Bakry-Émery Ricci flow on finite graphs. Our main result is the local existence and uniqueness of solutions to the Ricci flow. We prove the long-time convergence or finite-time blow up for the Bakry-Émery Ricci flow on finite trees and circles.

Figures (7)

• Figure 1: $T_3$ with boundary point $\delta T_3 = \{x_5, x_6, \cdots, x_{10}\}$.
• Figure 2: Bakry-Émery Ricci flow on $C_3$ with $m_0 = (2, 3, 4)$. In this cases, $\min_{x \in V}m(t, x)$ is decreasing.
• Figure 3: Bakry-Émery Ricci flow on $C_4$ with $m_0 = (2, 3, 4, 6)$. One can see that $\min_{x \in V}m(t, x)$ is decreasing. Moreover, for the vertex with the smallest initial weight, after a period of time, the weight is no longer the smallest.
• Figure 4: Bakry-Émery Ricci flow on $C_5$ with $m_0 = (2, \frac{1}{3}, 3, 1, \frac{5}{2})$.
• Figure 5:

Theorems & Definitions (25)

• Theorem 1.1
• Definition 2.1
• Remark 2.2
• Proposition 2.3
• Theorem 2.4
• proof
• Theorem 2.5
• Theorem 3.1
• proof
• Proposition 3.2