Ricci-flat graphs with girth at least five

TL;DR

This paper classifies all connected Ricci-flat graphs with girth at least five, identifying specific infinite and finite graphs, and constructs additional Ricci-flat graphs with smaller girth using Lie algebra root systems.

Contribution

It provides a complete classification of Ricci-flat graphs with girth ≥ 5 and introduces new Ricci-flat graphs with girth 3 or 4 via Lie algebra root systems.

Findings

Classified all Ricci-flat connected graphs with girth ≥ 5.

Identified specific graphs: infinite path, cycles, dodecahedral, Petersen, and half-dodecahedral.

Constructed Ricci-flat graphs with girth 3 or 4 using Lie algebra root systems.

Abstract

A graph is called Ricci-flat if its Ricci-curvatures vanish on all edges. Here we use the definition of Ricci-cruvature on graphs given in [Lin-Lu-Yau, Tohoku Math., 2011], which is a variation of [Ollivier, J. Funct. Math., 2009]. In this paper, we classified all Ricci-flat connected graphs with girth at least five: they are the infinite path, cycle $C_n$ ($n\geq 6$), the dodecahedral graph, the Petersen graph, and the half-dodecahedral graph. We also construct many Ricci-flat graphs with girth 3 or 4 by using the root systems of simple Lie algebras.