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