Disproof of a conjecture on the main spectrum of generalized Bethe trees

TL;DR

This paper disproves a conjecture stating that generalized Bethe trees with an even number of levels have exactly as many main eigenvalues as levels, by providing counterexamples for all even levels greater than or equal to six.

Contribution

The paper presents the first counterexamples to the conjecture, showing that the number of main eigenvalues can be different from the number of levels in generalized Bethe trees.

Findings

Counterexamples for even k ≥ 6.

Disproof of the conjecture for all even levels ≥ 6.

Main eigenvalues can be fewer or more than the number of levels.

Abstract

An eigenvalue of the adjacency matrix of a graph is said to be main if the all-ones vector is not orthogonal to its associated eigenspace. A generalized Bethe tree with $k$ levels is a rooted tree in which vertices at the same level have the same degree. Fran\c{c}a and Brondani [On the main spectrum of generalized Bethe trees, Linear Algebra Appl., 628 (2021) 56-71] recently conjectured that any generalized Bethe tree with $k$ levels has exactly $k$ main eigenvalues whenever $k$ is even. We disprove the conjecture by constructing a family of counterexamples for even integers $k\ge 6$.