The Multiplicity of Eigenvalues of A<sub>α</sub> of Trees
Zhibin Du
article (Uluslararası, Hakemli) | Yazar(lar): Zhibin Du | Yıl: 2026 | Dergi/Kurum: ANNALS OF COMBINATORICS | İndeks: WOS.SCI | DOI: 10.1007/s00026-025-00796-5 | Anahtar Kelimeler: Multiplicity of eigenvalues, A(alpha)matrix, Trees, Parter-Wiener Theorem, Downer branch — Özet: Let T be a tree on n vertices. For alpha is
Let T be a tree on n vertices. For alpha is an element of[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 mA(alpha) (T),theta as an eigenvalue of A(alpha) (T). When alpha=1/2 (aiming to the Laplacian matrix of T), it is known in Akbari et al. (Linear Algebra Appl 586:262-273, 2020) that mA(alpha) (T),theta <= n-3/2, and when alpha=0 (aiming to the adjacency matrix of T), the upper bound of mA(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 mA(alpha(T),theta) for any alpha is an element of[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.