The least $A_{\alpha}-$eigenvalue of graphs under perturbation

Abstract:

      Let $G$ be a simple undirected graph. For any real number $\alpha \in[0,1]$, Nikiforov defined the $A_{\alpha}$-matrix of $G$ as $A_{\alpha}(G)=\alpha D(G)+(1-\alpha)A(G)$, where $A(G)$ and $D(G)$ are the adjacency matrix and the degree diagonal matrix of $G$, respectively. In this paper, we investigate how the least eigenvalue of $A_\alpha(G)$ changes when the graph $G$ is perturbed by deleting a vertex, subdividing edges or moving edges, respectively.