  In 2021, Hibi et. al. studied lattice points in $\mathbb{N}^2$ that appear as
$(\depth R/I,\dim R/I)$ when $I$ is the edge ideal of a graph on $n$ vertices,
and showed these points lie between two convex polytopes. When restricting to
the class of Cameron--Walker graphs, they showed that these pairs do not form a
convex lattice polytope. In this paper, for the edge ideal $I$ of a
Cameron--Walker graph on $n$ vertices, we find how many points in
$\mathbb{N}^2$ appear as $(\depth(R/I),\dim(R/I))$, and how many points in
$\mathbb{N}^4$ appear as $(\depth(R/I),\reg(R/I),\dim(R/I),\degh(R/I)).$
