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. França and Brondani (2021) [4] 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≥6.
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