  Let $G$ be a connected graph of order $n$ with domination number $\gamma(G)$.
Wang, Yan, Fang, Geng and Tian [Linear Algebra Appl. 607 (2020), 307-318]
showed that for any Laplacian eigenvalue $\lambda$ of $G$ with multiplicity
$m_G(\lambda)$, it holds that $\gamma(G)\leq n-m_G(\lambda)$. Using techniques
from the theory of star sets, in this work we prove that the same bound holds
when $\lambda$ is an arbitrary adjacency eigenvalue of a non-regular graph, and
we characterize the cases of equality. Moreover, we show a result that gives a
relationship between start sets and the $p$-domination number, and we apply it
to extend the aforementioned spectral bound to the $p$-domination number using
the adjacency and Laplacian eigenvalue multiplicities.
