Graphs with the minimum spectral radius for given independence number

Let $G_{n,\alpha }$ be the set of connected graphs with order n and independence number α. The graph with the minimum spectral radius among $G_{n,\alpha }$ is called the minimizer graph. Stevanović in the classical book [Spectral Radius of Graphs, Academic Press, Amsterdam, 2015] pointed out that determining the minimizer graph in $G_{n,\alpha }$ appears to be a tough problem. Recently, Lou and Guo (2022) [14] proved that the minimizer graph in $G_{n,\alpha }$ 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 a constructing theorem for it. Thus, theoretically we determine the minimizer graphs in $G_{n,\alpha }$ along with their spectral radius for any given $\alpha \ge \lceil \frac{n}{2}\rceil$. As an application, we determine all the minimizer graphs in $G_{n,\alpha }$ for $\alpha =n-5,n-6$ along with their spectral radius.