Disproof of a conjecture on the minimum spectral radius and the domination number

Let $G_{n,\gamma }$ be the set of all connected graphs on $n$ vertices with domination number $\gamma$. A graph is called a minimizer graph if it attains the minimum spectral radius among $G_{n,\gamma }$. Very recently, Liu, Li and Xie [Linear Algebra and its Applications 673 (2023) 233–258] proved that the minimizer graph over all graphs in ${\mathbb{G}}_{n,\gamma }$ must be a tree. Moreover, they determined the minimizer graph among $G_{n,\lfloor \frac{n}{2}\rfloor }$ for even $n$, and posed the conjecture on the minimizer graph among $G_{n,\lfloor \frac{n}{2}\rfloor }$ for odd $n$. In this paper, we disprove the conjecture and completely determine the unique minimizer graph among $G_{n,\lfloor \frac{n}{2}\rfloor }$ for odd $n$.