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Graph bundles and Ricci-flatness

Wenbo Li, Shiping Liu

TL;DR

This work investigates graph bundles as the discrete analogue of fiber bundles and uses them to explore S-Ricci flat graphs beyond Abelian Cayley graphs. It defines graph bundles and discrete vector bundles, establishes when a non-trivial bundle cannot be isomorphic to a base-fiber Cartesian product, and analyzes vertex-transitivity via path projections and loop counts. The authors show that non-trivial bundles over vertex-transitive bases and fibers break vertex-transitivity in the discrete-vector-bundle setting, and they develop a systematic method to construct S-Ricci flat graphs through bundles, including explicit non-Abelian Cayley examples. They further identify conditions under which bundle graphs are locally Abelian, notably balance on 4-cycles, and provide concrete constructions and examples that illuminate the landscape of discrete Ricci-flat graphs and their relation to Cayley graphs.

Abstract

We develop a systematical way of constructing S-Ricci flat graphs which are not Abelian Cayley via graph bundle with explicit examples. For this purpose, we prove that, with some natural constrains, a non-trivial graph bundle can not be isomorphic (as graphs) to the product of the base graph and fiber graph. It stands in clear contrast to the continuous case.

Graph bundles and Ricci-flatness

TL;DR

This work investigates graph bundles as the discrete analogue of fiber bundles and uses them to explore S-Ricci flat graphs beyond Abelian Cayley graphs. It defines graph bundles and discrete vector bundles, establishes when a non-trivial bundle cannot be isomorphic to a base-fiber Cartesian product, and analyzes vertex-transitivity via path projections and loop counts. The authors show that non-trivial bundles over vertex-transitive bases and fibers break vertex-transitivity in the discrete-vector-bundle setting, and they develop a systematic method to construct S-Ricci flat graphs through bundles, including explicit non-Abelian Cayley examples. They further identify conditions under which bundle graphs are locally Abelian, notably balance on 4-cycles, and provide concrete constructions and examples that illuminate the landscape of discrete Ricci-flat graphs and their relation to Cayley graphs.

Abstract

We develop a systematical way of constructing S-Ricci flat graphs which are not Abelian Cayley via graph bundle with explicit examples. For this purpose, we prove that, with some natural constrains, a non-trivial graph bundle can not be isomorphic (as graphs) to the product of the base graph and fiber graph. It stands in clear contrast to the continuous case.
Paper Structure (5 sections, 7 theorems, 53 equations, 1 figure)

This paper contains 5 sections, 7 theorems, 53 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Graph bundles and discrete vector bundles
  3. Graph bundles and vertex transitivity
  4. Examples of non-trivial graph bundles
  5. Locally Abelian graphs

Key Result

Theorem 1

Let $G\times_\phi F$ be a graph bundle. The connection $\phi$ is trivial if and only if for any loop $\{x_{i}\}_{i=0}^N$ in $G$, where the product is defined by multiplication on the left.

Figures (1)

  • Figure 1: Graph Bundle $\mathbf{Z}_4\times_\phi K_4$

Theorems & Definitions (29)

  • Definition 1: Graph bundle
  • Definition 2: Discrete vector bundle
  • Definition 3: Equivalence of bundles
  • Remark 1
  • Remark 2
  • Definition 4: Paths
  • Theorem 1
  • proof
  • Theorem 2
  • Remark 3
  • ...and 19 more