Ricci-flat \(5\)-regular graphs (Q2694956)

A graph is Ricci-flat if the Ricci curvature vanishes on all edges. It is noticeable that the Ricci flat graphs under Lin-Lu-Yau's definition are different from the Ricci flat graphs defined in 1996 by Ch/ng and Yau Here, authors studied Ricci-flat 5-regular graphs that are not of the Cartesian product type The notion of Ricci curvature of Riemannian manifolds in differential geometry has been extended to other metric spaces such as graphs. The Ollivier-Ricci curvature between two vertices of a graph can be seen as a measure of how closely connected the neighbors of the vertices are compared to the distance between them. A Ricci-flat graph is then a graph in which each edge has curvature 0. There has been previous work in classifying Ricci-flat graphs under different definitions of Ricci curvature, notably graphs with large girth and small degrees under the definition of Lin-Lu-Yau, which is a modification of Ollivier's definition of Ricci curvature Here,continue the effort of classifying Ricci-flat graphs and study specifically Ricci-flat 5-regular graphs under the definition of Lin-Lu-Yau. When the symmetry condition for Ricci-flat graphs is not imposed, the possible cases for the construction of the graph grow enormously. The main difficulty of such a classification lies in the lack of leverageable symmetries. In Section 4, attention to symmetric graphs and found that Ricci-flat 5-regular symmetric graph must be isomorphic to a 5-regular symmetric graph of order 72. The paper contains interesting results .Useful to researchers working in the field of geometry and graph theory.