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

TL;DR

This paper explores the relationship between regularity and v-number of edge ideals of graphs, establishing bounds and explicitly characterizing pairs for special graph classes like whisker and Cameron-Walker graphs.

Contribution

It introduces the set of lattice points from regularity and v-number pairs, provides bounds, and characterizes these pairs for specific graph classes, also proposing a conjecture for chordal graphs.

Findings

Established bounds for the set of lattice points from regularity and v-number pairs.

Explicitly characterized pairs for whisker and Cameron-Walker graphs.

Proposed a conjecture for connected chordal graphs.

Abstract

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 $(\mathrm{reg}(R/I(G)), \mathrm{v}(I(G)))$, where $G$ ranges over all connected graphs on $n$ vertices, and we denote this set by $\mathcal{RV}(n)$. Here `$\mathrm{reg}$' denotes the (Castelnuovo-Mumford) regularity and `$\mathrm{v}$' denotes the $\mathrm{v}$-number. We establish general bounds for $\mathcal{RV}(n)$ by identifying two sets $A(n)$ and $B(n)$ satisfying $A(n)\subseteq \mathcal{RV}(n)\subseteq B(n)$. Furthermore, we explicitly determine the subsets of $\mathcal{RV}(n)$ consisting of all possible pairs $(\mathrm{reg}(R/I(G)), \mathrm{v}(I(G)))$ arising from whisker graphs and Cameron-Walker graphs on $n$ vertices. Finally, we propose a conjecture on the subset of $\mathcal{RV}(n)$ arising from connected chordal graphs.