Computing the characteristic polynomials of rooted trees and the energies of Bethe trees

Abstract

The mathematical problem of computing the characteristic polynomial of a tree is quite an old one, hence there are many known recursive formulae that deal with this matter. One of these methods is based on assigning a rational function to each vertex in a bottom-up manner, as described by Jacobs and Trevisan [Congr. Numer. 134 (1998) 139-145]. We improve the existing results by giving a shorter and more concise proof of an extension of this method. Furthermore, we implement the aforementioned formula in order to investigate the spectral properties of balanced trees, with a special focus on the Bethe trees, as already done so by Heydari and Taeri [Linear Algebra Appl. 429 (2008) 1744-1757], as well as Robbiano and Trevisan [Comput. Math. with Appl. 59 (2010) 3039-3044]. We improve those results by demonstrating a quicker way to prove the formula that computes the characteristic polynomial of a balanced tree. Finally, we use a technique inspired by Damnjanović et al. [arXiv:/2210.08337 (2022)] in order to obtain a fully computed expression for the energy of any Bethe tree, thereby extending the existing results.