Representations of regular trees and invariants of AR-components for generalized Kronecker quivers

We investigate the generalized Kronecker algebra \(\mathcal{K}_r = k\Gamma_r\) with \(r \geq 3\) arrows. Given a regular component \(\mathcal{C}\) of the Auslander-Reiten quiver of \(\mathcal{K}_r\), we show that the quasi-rank \(rk(\mathcal{C}) \in \mathbb{Z}_{\leq 1}\) can be described almost exactly as the distance \(\mathcal{W}(\mathcal{C}) \in \mathbb{N}_0\) between two non-intersecting cones in \(\mathcal{C}\), given by modules with the equal images and the equal kernels property; more precisley, we show that the two numbers are linked by the inequality \[ -\mathcal{W}(\mathcal{C}) \leq rk(\mathcal{C}) \leq - \mathcal{W}(\mathcal{C}) + 3.\] Utilizing covering theory, we construct for each \(n \in \mathbb{N}_0\) a bijection \(\varphi_n\) between the field \(k\) and \(\{ \mathcal{C} \mid \mathcal{C} \ \text{regular component}, \ \mathcal{W}(\mathcal{C}) = n \}\). As a consequence, we get new results about the number of regular components of a fixed quasi-rank.