Discrete Einstein metrics on unicyclic graphs
arXiv:2607.14748
The paper extends the study of discrete Einstein metrics defined via Lin‑Lu‑Yau curvature from trees to unicyclic graphs, identifying when the tree‑based solution persists, providing closed‑form results for cycles and regular suns, and analyzing existence and uniqueness regimes.
Abstract
In earlier work with Cheng and Hua we showed that on a finite tree the discrete Einstein metrics of the Lin--Lu--Yau curvature are the Perron eigenvector of an edge-indexed Ricci matrix. We extend this theory to unicyclic graphs. We determine exactly when the tree picture persists -- the balanced regime, where the spectrum becomes periodic rather than Dirichlet-type -- and compute it in closed form for bare cycles and for regular suns (cycles with pendant leaves); for a single decorated vertex on a long cycle it persists up to an explicit golden-ratio threshold. Beyond this regime the problem is piecewise-linear, and phenomena impossible on a tree appear: the Einstein metric can be non-unique, or absent -- a triangle with a pendant leaf carries none. For the regular suns we prove that it exists and is unique.