On the Number of Arrows of Cluster Quivers

TL;DR

This paper investigates the distribution of the number of arrows in cluster quivers of finite mutation type, establishing bounds, continuity properties, and conditions for infinite arrow counts, thereby advancing understanding of their combinatorial structure.

Contribution

It introduces the distribution set of arrow counts, determines bounds, proves continuity except in special cases, and characterizes when infinite arrows occur in mutation classes.

Findings

Distribution set of arrow counts is continuous except for exceptional cases.

Maximum number of arrows in infinite mutation type quivers can be infinite.

Existence of complete walks in the mutation classes is established.

Abstract

Let $\tilde{Q}$ (resp. $Q$) be an extended exchange (resp. exchange) cluster quiver of finite mutation type. We introduce the distribution set of the number of arrows for $Mut[\tilde{Q}]$ (resp. $Mut[Q]$), give the maximum and minimum numbers of the distribution set and establish the existence of an extended complete walk (resp. a complete walk). As a consequence, we prove that the distribution set for $Mut[\tilde{Q}]$ (resp. $Mut[Q]$) is continuous except the exceptional cases. In case of cluster quivers $Q_{inf}$ of infinite mutation type, the number of arrows does not present a continuous distribution. Besides, we show that the maximal number of arrows of quivers in $Mut[Q_{inf}]$ is infinite if and only if the maximal number of arrows between any two vertices of a quiver in $Mut[Q_{inf}]$ is infinite.