Most A$_α$-eigenvalues of a tree are small

Source abstract

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