Ricci-flat graphs with girth four

Abstract

Lin-Lu-Yau introduced an interesting notion of Ricci curvature for graphs and obtained a complete characterization for all Ricci-flat graphs with girth at least five [1]. In this paper, we propose a concrete approach to construct an infinite family of distinct Ricci-flat graphs of girth four with edge-disjoint 4-cycles and completely characterize all Ricci-flat graphs of girth four with vertexdisjoint 4-cycles.

Ricci-flat graphs with maximum degree at most 4

“…We have determined the Ricci-flat graphs in class G that contains an edge with endpoint degree {2, 3} in Theorem 2. In this section, we continue with endpoint degree (3, 3) and (3,4). To determine the former case (see Theorem 5), we will need the Theorem 3 .…”

“…In previous sections, we have finished all cases when an edge has endpoint degrees (2, 2), (2, 3), (3,3), (3,4). In this section, we consider the Ricci-flat graphs that contain edges with endpoint degrees (2, 4), (4,4), we first classify these that contain a copy of C 3 .…”

“…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. While the fact is there are infinitely many Ricci-flat graphs with girth three or four.…”

“…Vertices in Γ(x) are labeled by first d(x) positive integers, vertices in Γ(y) are labeled by the succeeding integers. Note in [3], the authors also analyzed the local structures of edge with Ricci curvature 0, the difference is that their conclusions are based on that the the graph has girth 4 and the 4-cycles in the graph are mutually vertex-disjoint. Our results on these local structures can be applied to graphs without any restriction.…”

“…Let G be a Ricci-flat graph with maximum vertex degree 4, assume G contains no edges with endpoint degree(3,4), and all C 4 s of G are vertex-disjoint. Then G is isomorphic to G 8 .…”