Graphs with the minimum spectral radius for given independence number

Let $\mathbb{G}_{n,\alpha}$ be the set of connected graphs with order $n$ and independence number $\alpha$. Given $k=n-\alpha$, the graph with minimum spectral radius among $\mathbb{G}_{n,\alpha}$ is called the minimizer graph. Stevanovi\'{c} in the classical book [D. Stevanovi\'{c}, Spectral Radius of Graphs, Academic Press, Amsterdam, 2015.] pointed that determining minimizer graph in $\mathbb{G}_{n,\alpha}$ appears to be a tough problem on page $96$. Very recently, Lou and Guo in \cite{Lou} proved that the minimizer graph of $\mathbb{G}_{n,\alpha}$ must be a tree if $\alpha\ge\lceil\frac{n}{2}\rceil$. In this paper, we further give the structural features for the minimizer graph in detail, and then provide of a constructing theorem for it. Thus, theoretically we completely determine the minimizer graphs in $\mathbb{G}_{n,\alpha}$ along with their spectral radius for any given $k=n-\alpha\le \frac{n}{2}$. As an application, we determine all the minimizer graphs in $\mathbb{G}_{n,\alpha}$ for $\alpha=n-1,n-2,n-3,n-4,n-5,n-6$ along with their spectral radii, the first four results are known in \cite{Xu,Lou} and the last two are new.