On the Number of Arrows of Cluster Quivers

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.