Ollivier Ricci curvature of Cayley graphs for dihedral groups, generalized quaternion groups, and cyclic groups

TL;DR

This paper calculates the Ricci curvatures of Cayley graphs for dihedral, quaternion, and cyclic groups using a specific formulation, providing explicit curvature values for various generating sets.

Contribution

It extends Ricci curvature calculations to Cayley graphs of dihedral, quaternion, and cyclic groups with small generating sets, offering explicit curvature values.

Findings

Ricci curvatures for dihedral group Cayley graphs are fully determined.

Explicit Ricci curvature values for quaternion group Cayley graphs are provided.

Curvatures for cyclic group Cayley graphs with specific generators are calculated.

Abstract

Lin, Lu, and Yau formulated the Ricci curvature of edges in simple undirected graphs[2]. Using their formulations, we calculate the Ricci curvatures of Cayley graphs for the dihedral groups, the general quaternion groups, and cyclic groups with some generating sets that are chosen so that their cardinal numbers are less than or equal to four. For the dihedral group and the general quaternion group, we obtained the Ricci curvatures of all edges of the Cayley graph with generator sets consisting of the four elements that are the two generators defining each group and their inverses elements.For the cyclic group (Z/nZ, +), we have the Ricci curvatures of edges of the Cayley graph generating by S_{1, k} = {+1, -1, +k, -k}.