Sharp bounds on the $A_{\alpha }$-index of graphs in terms of the independence number

Abstract

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 $A_{\alpha }$-matrix: $A_{\alpha }(G)=\alpha D(G)+(1-\alpha )A(G),$ where $\alpha \in [0,1]$. The largest eigenvalue of this novel matrix is called the $A_{\alpha }$-index of $G$. In this paper, we characterize the graphs with minimum $A_{\alpha }$-index among $n$-vertex graphs with independence number $i$ for $\alpha \in [0,1)$, 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 $\alpha \in [0,\frac{3}{4}].$ Furthermore, we determine the unique graph (resp. tree) on $n$ vertices with given independence number having the maximum $A_{\alpha }$-index with $\alpha \in [0,1)$, whereas for the $n$-vertex bipartite graphs with given independence number, we characterize the unique graph having the maximum $A_{\alpha }$-index with $\alpha \in [\frac{1}{2},1).$

21 pages; 4 figures; It is accepted by Acta Math. Appl. Sin. Engl. Ser