bibkey: huangstoicaxiazhong2020einstein
authors: An Huang; Bogdan Stoica; Xuyang Xia; Xiao Zhong
year: 2020
title: "Bounds on the Ricci curvature and solutions to the Einstein equations for weighted graphs"
doi: null
url: "https://arxiv.org/abs/2006.06716v1"
claim: "The paper defines Ricci curvature and an action for weighted graphs, derives the equations of motion on regular trees, proves uniqueness of the constant solution under constant boundary conditions, constructs solutions on the line and on trees of even degree, and conjectures that the tree equations of motion on the 3-regular tree have no positive non-constant solution."
strata_touched:
  - D5/S3/Geometry/DiscreteGravity/TreeEquationsNonconstantRefutation
license: citation-only
triage: anchor

Huang, Stoica, Xia and Zhong, Bounds on the Ricci curvature and solutions to the Einstein equations for weighted graphs

A. Huang, B. Stoica, X. Xia and X. Zhong, arXiv:2006.06716v1 (2020); arXiv lists v1 only, with no journal reference. Quotations are from the arXiv v1 TeX source.

For a graph with positive edge lengths, $P_{ij}$ is the length of a shortest path from $i$ to $j$, $d_i={\sum }_{i^{\prime }\sim i}{P}_{i{i}^{\prime }}^{-2}$ and $c_i={\sum }_{i^{\prime }\sim i}{P}_{i{i}^{\prime }}^{-1}$. $T_q$ is the infinite tree in which every vertex has degree $q+1$. On $T_q$ the tree equation of motion on an edge $ij$ is $$\frac{1}{P_{ij}}(\frac{c_i^2}{d_i^2}+\frac{c_j^2}{d_j^2})-\frac{c_i}{d_i}-\frac{c_j}{d_j}=0,$$ and the setting with all edges equal is the constant solution. The paper closes with two conjectures, the second of which reads:

There exists no non-constant solution to the tEoM for $T_2$ with all edge lengths positive.

Verified locator

Source: https://arxiv.org/abs/2006.06716v1

• URL: https://arxiv.org/abs/2006.06716v1. The e-print ms.tex has 1996 lines and md5 de7aa04a62501a159118abdc2fbc2691. Locations:
  • $P_{ij}$, l. 235; $d_i$ and $c_i$, l. 255–257;
  • $T_q$, l. 542; the tree equation of motion and the constant solution, l. 549–556;
  • the two conjectures, l. 1924–1935.