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

Iwao Mizukai, Akifumi Sako

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}.

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

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}.
Paper Structure (25 sections, 18 theorems, 153 equations, 27 figures, 14 tables)

This paper contains 25 sections, 18 theorems, 153 equations, 27 figures, 14 tables.

Table of Contents

  1. Introduction
  2. Preparations
  3. Ricci curvature of graph
  4. Cayley graph
  5. The Ricci curvature of the Cayley graph of dihedral groups
  6. The Ricci curvature of the Cayley graph for generalized quaternion groups
  7. generalized quaternion group
  8. Ricci curvature of Cayley graph for generalized quaternion group $Q_{8}$ with the generating set $S = \{ \sigma, \tau, \sigma^{-1}, \tau^{-1} \}$
  9. The Ricci curvature of the Cayley graph for cyclic groups
  10. Cyclic group and Cayley graph
  11. The Ricci curvature of the Cayley graph for the cyclic group $\Gamma (\mathbb{Z}/n\mathbb{Z}, S_{1})$ with the generating set $S_{1}=\{ +1, -1 \}$
  12. The Ricci curvature of the Cayley graph for the cyclic group $\Gamma (\mathbb{Z}/n\mathbb{Z}, S_{1, 2})$ with the generating set $S_{1, 2}=\{ +1,+2 -1, -2 \}$
  13. The Ricci curvature of the Cayley graph for the cyclic group $\Gamma (\mathbb{Z}/n\mathbb{Z}, S_{1, 3})$ with the generating set $S_{1, 3}=\{ +1,+3, -1, -3 \}$
  14. The Ricci curvature of the Cayley graph for the cyclic group $\Gamma (\mathbb{Z}/n\mathbb{Z}, S_{1, 4})$ with the generating set $S_{1, 4}=\{ +1, +4, -1, -4 \}$
  15. The Ricci curvature of the Cayley graph for the cyclic group $\Gamma (\mathbb{Z}/n\mathbb{Z}, S_{1, 5})$ with the generating set $S_{1, 5}=\{ +1, +5, -1, -5 \}$
  16. ...and 10 more sections

Key Result

Theorem 1

olliLin Let $G=(V, E)$ be a connected graph. For $\mu, \nu \in P(V)$,

Figures (27)

  • Figure 1: Type A in the Cayley graph of $D_{3}$
  • Figure 2: Type B in the Cayley graph of $D_{3}$
  • Figure 3: Type A in the Cayley graph of $D_{n}$
  • Figure 4: Type B in the Cayley graph of $D_{n}$
  • Figure 5: Type A in the Cayley graph of $Q_{4m}$
  • ...and 22 more figures

Theorems & Definitions (39)

  • Definition 1
  • Definition 2
  • Definition 3
  • Theorem 1
  • Definition 4
  • Definition 5
  • Definition 6
  • Definition 7
  • Definition 8
  • Proposition 1
  • ...and 29 more