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 $\gamma \ge \lceil \frac{n}{2}\rceil$, and we provide several characterizations of such $A_{\alpha }$-minimizer graphs. We then specifically characterize the $A_{\alpha }$-minimizer graphs for the case $\gamma =\lceil \frac{n}{2}\rceil +1$ when $n\ge 9$. Furthermore, we obtain a structural characterization for the $A_{\alpha }$-minimizer graph when $\gamma =n-c$, where $c\ge 4$ is an integer.