The minimum spectral radius of graphs with a given domination number

Abstract: Let $\mathbb{G}{n,\gamma}$ be the set of simple and connected graphs on $n$ vertices and with domination number $\gamma$. The graph with minimum spectral radius among $\mathbb{G}{n,\gamma}$ is called the minimizer graph. In this paper, we first prove that the minimizer graph of $\mathbb{G}{n,\gamma}$ must be a tree. Moreover, for $\gamma\in{1,2,3,\lceil\frac{n}{3}\rceil,\lfloor\frac{n}{2}\rfloor}$, we characterize all minimizer graphs in $\mathbb{G}{n,\gamma}$.