Ricci-flat graphs with maximum degree at most 4

Abstract

A graph is called Ricci-flat if its Ricci curvatures vanish on all edges, here the definition of Ricci curvature on graphs was given by Lin-Lu-Yau [5]. The authors in [4] and [2] obtained a complete characterization for all Ricci-flat graphs with girth at least five. In this paper, we completely determined all Ricci-flat graphs with maximum degree at most 4.

Graphs with nonnegative Ricci curvature and maximum degree at most 3

“…Then G is isomorphic to a finite path, a simple cycle, a prism graph, a Möbius ladder, a particular graph as shown in Figure 1, or a quasi-ladder graph as shown in Figure 2. Bai, Lu and Yau [BLY21] classified all Ricci-flat graphs with maximum degree at most 4. Compared with our results, they require a more stringent curvature condition but relax the maximum degree condition from 3 to 4, which causes a huge increase in the complexity of the problem.…”