Most A${}_{\alpha }$-eigenvalues of a tree are small

We study the distribution of $A_{\alpha }$-eigenvalues of a tree. We extend the works \cite{Jacobs2021} and \cite{SIN2020} by proving that the number of $A_{\alpha }$-eigenvalues, for $0\le \alpha \le \frac{1}{2}$, less than or equal to the average degree $d_{\alpha }=\alpha (2-2\mathrm{/}n)$ is, at least, $\lceil \frac{n}{2}\rceil$ for a tree with $n$ vertices. Several counterexamples for $\frac{1}{2}<\alpha \le 1$ are exhibited. We also derive the same result for other well-known families such as the Deformed Laplacian and the matrix $B_{\beta }$.