The Multiplicity of $A_{\alpha}$-eigenvalues of Graphs

Abstract

For a graph $G$ and real number $\alpha\in [0,1]$, the $A_{\alpha}$-matrix of $G$ is defined as $A_{\alpha}(G)=\alpha D(G)+(1-\alpha)A(G),$ where $A(G)$ is the adjacency matrix of $G$ and $D(G)$ is the diagonal matrix of the vertex degrees of $G$. In this paper, the largest multiplicity of the $A_{\alpha}$-eigenvalues of a broom tree is considered, and all graphs with an $A_{\alpha}$-eigenvalue of multiplicity at least $n-2$ are characterized.