Characterizing $A_\alpha$-minimizer graphs: given order and independence number

For a given graph $G$, let $A(G)$, $Q(G)$, and $D(G)$ denote the adjacency matrix, signless Laplacian matrix, and diagonal degree matrix of $G$, respectively. The $A_\alpha(G)$ matrix, proposed by Nikiforov, is defined as $A_\alpha(G)=\alpha D(G)+(1 - \alpha)A(G)$, where $\alpha\in[0,1]$. This matrix captures the gradual transition from $A(G)$ to $Q(G)$. Let $\mathcal{G}_{n,\gamma}$ denote the family of all connected graphs with $n$ vertices and independence number $\gamma$. A graph in $\mathcal{G}_{n,\gamma}$ is referred to as an $A_\alpha$-minimizer graph if it achieves the minimum $A_\alpha$ spectral radius. In this paper, we first demonstrate that the $A_\alpha$-minimizer graph in $\mathcal{G}_{n,\gamma}$ must be a tree when $\gamma\geq\left\lceil\frac{n}{2}\right\rceil$, and we provide several characterizations of such $A_\alpha$-minimizer graphs. We then specifically characterize the $A_\alpha$-minimizer graphs for the case $\gamma = \left\lceil\frac{n}{2}\right\rceil + 1$ when $n\geq 9$. Furthermore, we obtain a structural characterization for the $A_\alpha$-minimizer graph when $\gamma=n - c$, where $c\geq4$ is an integer.