Year: 2026
Type: preprint
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 $B_β$.
Source: arXiv (Cornell University)
Authors: Barturén, Elvia P., Oliveira, Elismar R.
Language: English
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Award: G2013447816
OpenAlex ID: W7214901786