Discrete Einstein metrics on unicyclic graphs
Abstract
In earlier work with Cheng and Hua we showed that on a finite tree the Lin--Lu--Yau discrete Einstein equation is a single linear eigenvalue problem: the metric is the Perron eigenvector of an edge-indexed Ricci matrix. We extend this theory to unicyclic graphs. For geodesic cycle weightings of girth at least four, we characterize exactly when the same local matrix computes the true curvature; this is the balanced regime. Schur complementation of the pendant trees then reduces the eigenvalue problem to a periodic transfer equation on the cycle. We obtain closed formulas for bare cycles and regular suns, prove a sharp maximum-degree bound, and show that the spectral top of a fixed localized defect on a growing cycle converges to the top bound state of the corresponding infinite-line operator. This yields an explicit golden-ratio threshold for balance. Outside the balanced regime the active shortest paths and optimal transport constraints must also be tracked, and phenomena impossible on trees occur: the bare four-cycle has a one-parameter family of Einstein metrics, whereas a triangle with one pendant leaf has none. For every regular sun we construct the symmetric Einstein metric and prove that it is the unique Einstein metric, up to scale, among all positive weightings. As long as a normalized Ricci-flow trajectory remains in the balanced cone, its projective dynamics is generated by the fixed Ricci matrix.
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BibTeX
@misc{bai2026discrete,
title = {{Discrete Einstein metrics on unicyclic graphs}},
author = {Shuliang Bai},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2607.14748},
eprint = {2607.14748},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.14748},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Discrete Einstein metrics on unicyclic graphs
AU - Shuliang Bai
PY - 2026
DA - 2026/09/18
PB - arXiv
DO - 10.48550/arxiv.2607.14748
UR - https://arxiv.org/abs/2607.14748
AB - In earlier work with Cheng and Hua we showed that on a finite tree the Lin--Lu--Yau discrete Einstein equation is a single linear eigenvalue problem: the metric is the Perron eigenvector of an edge-indexed Ricci matrix. We extend this theory to unicyclic graphs. For geodesic cycle weightings of girth at least four, we characterize exactly when the same local matrix computes the true curvature; this is the balanced regime. Schur complementation of the pendant trees then reduces the eigenvalue problem to a periodic transfer equation on the cycle. We obtain closed formulas for bare cycles and regular suns, prove a sharp maximum-degree bound, and show that the spectral top of a fixed localized defect on a growing cycle converges to the top bound state of the corresponding infinite-line operator. This yields an explicit golden-ratio threshold for balance. Outside the balanced regime the active shortest paths and optimal transport constraints must also be tracked, and phenomena impossible on trees occur: the bare four-cycle has a one-parameter family of Einstein metrics, whereas a triangle with one pendant leaf has none. For every regular sun we construct the symmetric Einstein metric and prove that it is the unique Einstein metric, up to scale, among all positive weightings. As long as a normalized Ricci-flow trajectory remains in the balanced cone, its projective dynamics is generated by the fixed Ricci matrix.
ER -