| Περίληψη: |
Let T be a tree on n vertices. For α ∈ [ 0 , 1) , let A α (T) = α D (T) + (1 - α) A (T) , where D(T) is the degree diagonal matrix of T and A(T) is the adjacency matrix of T. Denote by m A α (T) , θ the multiplicity of θ as an eigenvalue of A α (T). When α = 1 2 (aiming to the Laplacian matrix of T), it is known in Akbari et al. (Linear Algebra Appl 586:262–273, 2020) that m A α (T) , θ ≤ n - 3 2 , and when α = 0 (aiming to the adjacency matrix of T), the upper bound of m A α (T) , θ for any fixed real number θ was recently obtained in Du and Huang (Linear Algebra Appl 654:56–68, 2022). In this paper, we establish an upper bound of m A α (T) , θ for any α ∈ [ 0 , 1) and any fixed real number θ , unifying and generalizing the above-mentioned results about the multiplicities of eigenvalues of the adjacency matrix and Laplacian matrix of trees. [ABSTRACT FROM AUTHOR] |