# Cluster algebras and symmetrizable matrices

## TL;DR

This paper explores the combinatorial properties of quasi-Cartan companions in acyclic skew-symmetrizable cluster algebras, revealing that their associated diagrams always contain an admissible cut of edges.

## Contribution

It demonstrates that diagrams of skew-symmetrizable matrices in acyclic cluster algebras have an admissible cut, advancing understanding of their combinatorial structure.

## Key findings

- Diagrams have an admissible cut of edges.
- Properties of quasi-Cartan companions are characterized.
- Results apply to all acyclic skew-symmetrizable cluster algebras.

## Abstract

In this paper, we study combinatorial properties of quasi-Cartan companions defined by the c-vectors of acyclic skew-symmetrizable cluster algebras. In particular, we show that the diagram of any skew-symmetrizable matrix associated with an acyclic cluster algebra has an admissible cut of edges.