Deleting Vertices and Interlacing of $A_\alpha$ Eigenvalues of a Graph
Received:September 06, 2021  Revised:December 23, 2021
Key Words: $A_{\alpha}$ eigenvalue   interlacing inequality   independence number   cover number   Hamiltonian properties   spanning tree
Fund Project:Supported by the National Natural Science Foundation of China (Grant No.12171089) and the Natural Science Foundation of Fujian Province (Grant No.2021J02048).
 Author Name Affiliation Hongzhang CHEN School of Mathematics and Statistics, Minnan Normal University, Fujian 363000, P. R. China Jianxi LI School of Mathematics and Statistics, Minnan Normal University, Fujian 363000, P. R. China
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Let $G$ be simple graph with vertex set $V$ and edge set $E$. In this paper, we establish an interlacing inequality between the $A_{\alpha}$ eigenvalues of $G$ and its subgraph $G-U$, where $U\subseteq V$. Moreover, as an application, this interlacing property can be used to deduce some $A_{\alpha}$ spectral conditions concerning the independence number, cover number, Hamiltonian property and spanning tree of a graph, respectively.