Lattice points arising from regularity and $\mathrm{v}$-number of Graphs: Whisker and Cameron-Walker

Let $G$ be a simple graph on $n$ vertices and $I(G)\subseteq R$ be its edge ideal. In this paper, we initiate the study of determining lattice points in ${\mathbb{N}}^2$ that appear as a pair $(\mathrm{r}\mathrm{e}\mathrm{g}(R\mathrm{/}I(G)),\mathrm{v}(I(G)))$, where $G$ ranges over all connected graphs on $n$ vertices, and we denote this set by $\mathcal{R}\mathcal{V}(n)$. Here $\mathrm{reg}$' denotes the (Castelnuovo-Mumford) regularity and $\mathrm{v}$’ denotes the $\mathrm{v}$-number. We establish general bounds for $\mathcal{R}\mathcal{V}(n)$ by identifying two sets $A(n)$ and $B(n)$ satisfying $A(n)\subseteq \mathcal{R}\mathcal{V}(n)\subseteq B(n)$. Furthermore, we explicitly determine the subsets of $\mathcal{R}\mathcal{V}(n)$ consisting of all possible pairs $(\mathrm{r}\mathrm{e}\mathrm{g}(R\mathrm{/}I(G)),\mathrm{v}(I(G)))$ arising from whisker graphs and Cameron-Walker graphs on $n$ vertices. Finally, we propose a conjecture on the subset of $\mathcal{R}\mathcal{V}(n)$ arising from connected chordal graphs.