Solsolitons associated with graphs

TL;DR

This paper introduces a method to associate graphs with certain positivity properties to families of Ricci soliton solvable Lie groups, classifies these solitons, and categorizes small graphs based on positivity.

Contribution

It provides a novel classification linking graph properties to geometric structures of Ricci solitons on Lie groups, with explicit examples and parameter families.

Findings

Classification of solsolitons up to isometry

Explicit examples of Ricci solitons with parameter families

Categorization of small graphs based on positivity

Abstract

We show how to associate with each graph with a certain property (positivity) a family of simply connected solvable Lie groups endowed with left-invariant Riemannian metrics that are Ricci solitons (called solsolitons). We classify them up to isometry, obtaining families depending on many parameters of explicit examples of Ricci solitons. A classification of graphs with up to 3 coherent components according to positivity is also given.