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 ${\mathbb{G}}_{n,\gamma }$.