  Let $R$ be a $G$ graded commutative ring and $M$ be a $G$-graded $R$-module.
The set of all graded second submodules of $M$ is denoted by $Spec_G^s(M)$ and
it is called the graded second spectrum of $M$. In this paper, we discuss
graded rings with Noetherian graded prime spectrum and obtain some conclusions.
In addition, we introduce the notion of the graded Zariski socle of graded
submodules and explore their properties. Using these conclusions and
properties, we also investigate $Spec_G^s(M)$ with the Zariski topology from
the viewpoint of being a Noetherian space and give some related outcomes.
