Sharp bounds on the $A_α$-index of graphs in terms of the independence number
Given a graph $G$, the adjacency matrix and degree diagonal matrix of $G$ are denoted by $A(G)$ and $D(G)$, respectively. In 2017, Nikiforov \cite{0007} proposed the $A_α$-matrix: $A_α(G)=αD(G)+(1-α)A(G),$ where $α\in [0, 1]$. The largest eigenvalue of this novel matrix is called the $A_α$-index of $G$. In this paper, we characterize the graphs with minimum $A_α$-index among $n$-vertex graphs with independence number $i$ for $α\in[0,1)$, where $i=1,\lfloor\frac{n}{2}\rfloor,\lceil\frac{n}{2}\rceil,{\lfloor\frac{n}{2}\rfloor+1},n-3,n-2,n-1,$ whereas for $i=2$ we consider the same problem for $α\in [0,\frac{3}{4}{]}.$ Furthermore, we determine the unique graph (resp. tree) on $n$ vertices with given independence number having the maximum $A_α$-index with $α\in[0,1)$, whereas for the $n$-vertex bipartite graphs with given independence number, we characterize the unique graph having the maximum $A_α$-index with $α\in[\frac{1}{2},1).$