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 $(\depth R/I,\dim R/I)$ 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 $(\depth(R/I),\reg(R/I),\dim(R/I),\degh(R/I)).$
Comment: 16 pages, 4 figures