Counting lattice points that appear as algebraic invariants of Cameron-Walker graphs

In 2021, Hibi et. al. studied lattice points in ${\mathbb{N}}^2$ that appear as $(\text{depth}R\mathrm{/}I,\dim R\mathrm{/}I)$ 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 $(\text{depth}(R\mathrm{/}I),\dim (R\mathrm{/}I))$, and how many points in ${\mathbb{N}}^4$ appear as $(\text{depth}(R\mathrm{/}I),\text{reg}(R\mathrm{/}I),\dim (R\mathrm{/}I),\text{degh}(R\mathrm{/}I))\mathrm{.}$