Discrete Einstein metrics on unicyclic graphs

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.