Ricci-flat cubic graphs with girth five

Abstract

We classify all connected, simple, 3-regular graphs with girth at least 5 that are Ricci-flat. We use the definition of Ricci curvature on graphs given in Lin-Lu-Yau, Tohoku Math., 2011, which is a variation of Ollivier, J. Funct. Anal., 2009. A graph is Ricci-flat, if it has vanishing Ricci curvature on all edges. We show, that the only Ricci-flat cubic graphs with girth at least 5 are the Petersen graph, the Triplex and the dodecahedral graph. This will correct the classification in [8] that misses the Triplex.

Graph Laplacians, Riemannian Manifolds and their Machine-Learning

“…One way to define Ricci-flatness is to modify the notion of Ricci curvature for metric spaces in the sense of [Oll] The classification was addressed in [LLY] (cf. also [CKLLLY,CKLLLY2,OSY]). For vertices x, y ∈ V , and any real value α ∈ [0, 1], define the probability distribution µ α…”

Ricci-flat graphs with maximum degree at most $4$

“…Thus one of d (3,4), d(3, 5) must be 2. By the symmetry of vertex 4 and vertex 5, wlog, let d(3, 5) = 2.…”

“…We have determined the Ricci-flat graphs in class G that contains an edge with endpoint degree {2, 3}. In this section, we continue with endpoint degree (3,3) and (3,4). To determine the former case (see Theorem 4), we will need the Theorem 3.…”

“…There have been works on classifying Ricci-flat graphs. At first, [8] and [3] classified all Ricci-flat graphs with girth at least five. Then the authors in [4] characterized all Ricci-flat graphs of girth four with vertex-disjoint 4-cycles, their results show that there are two such graphs.…”

Ricci-flat graphs with maximum degree at most 4

“…There have been works on classifying Ricci-flat graphs. At first, [4] and [2] classified all Ricci-flat graphs with girth at least five. Then the authors in [3] characterized all Ricci-flat graphs of girth four with vertex-disjoint 4-cycles, their results show that there are two such graphs.…”