Characterizing $A_σ$-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 $A_σ(G)$ matrix, proposed by Nikiforov, is defined as $A_σ(G)=σD(G)+(1 - σ)A(G)$, where $σ\in[0,1]$. This matrix captures the gradual transition from $A(G)$ to $Q(G)$. Let $\mathcal{G}_{n,α}$ denote the family of all connected graphs with $n$ vertices and independence number $α$. A graph in $\mathcal{G}_{n,α}$ is referred to as an $A_σ$-minimizer graph if it achieves the minimum $A_σ$ spectral radius. In this paper, we first demonstrate that the $A_σ$-minimizer graph in $\mathcal{G}_{n,α}$ must be a tree when $α\geq\left\lceil\frac{n}{2}\right\rceil$, and we provide several characterizations of such $A_σ$-minimizer graphs. We then specifically characterize the $A_σ$-minimizer graphs for the case $α= \left\lceil\frac{n}{2}\right\rceil + 1$. Furthermore, we obtain a structural characterization for the $A_σ$-minimizer graph when $α=n - c$, where $c\geq4$ is an integer. Finally, we identify 17 potential $A_σ$-minimizer graphs within $\mathcal{G}_{n,n - 4}$, thereby extending the results of Liu and Wang [9].