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 $n$.
Categories: math.CO