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

Prativa Biswas, Mousumi Mandal, Kamalesh Saha

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.

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

Abstract

Let be a simple graph on vertices and be its edge ideal. In this paper, we initiate the study of determining lattice points in that appear as a pair , where ranges over all connected graphs on vertices, and we denote this set by . Here `' denotes the (Castelnuovo-Mumford) regularity and `' denotes the -number. We establish general bounds for by identifying two sets and satisfying . Furthermore, we explicitly determine the subsets of consisting of all possible pairs arising from whisker graphs and Cameron-Walker graphs on vertices. Finally, we propose a conjecture on the subset of arising from connected chordal graphs.
Paper Structure (6 sections, 18 theorems, 47 equations, 5 figures)

This paper contains 6 sections, 18 theorems, 47 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. An approximation of the set $\mathcal{RV}(n)$
  4. The set $\mathcal{RV}_{W}(n)$: Restriction of $\mathcal{RV}(n)$ to whisker graphs
  5. The set $\mathcal{RV}_{CW}(n)$: Restriction of $\mathcal{RV}(n)$ to Cameron-Walker graphs
  6. Future Directions

Key Result

Theorem \ref{thm:RV(n)}

Let $A(n)=\{(r,v)\in \mathbb{N}^{2} \mid 1\leq r<\frac{n}{2}, 1\leq v\leq r-\lceil\frac{r}{n-2r}\rceil+1\}$ and $B(n)=\{(r,v)\in\mathbb{N}^2 \mid 1\leq r<\frac{n}{2}, 1\leq v<\frac{n}{2}\}$. Then for $n\geq 3$, we have

Figures (5)

  • Figure 1: A tree $G$ with $n-2r\geq r$ and $(\operatorname{reg}(G),\mathrm{v}(G))=(r,v)$
  • Figure 2: A chordal graph $G$ with $n-2r<r$ and $(\operatorname{reg}(G),\mathrm{v}(G))=(r,v)$
  • Figure 3: A whisker graph $W_G$ with $(\operatorname{reg}(W_G),\mathrm{v}(W_G))=(r,v)$
  • Figure 4: A Cameron-walker graph
  • Figure 5: A Cameron-walker graph $G$ with $(\operatorname{reg}(G),\mathrm{v}(G))=(r,v)$

Theorems & Definitions (37)

  • Theorem \ref{thm:RV(n)}
  • Theorem \ref{thm:wrv}
  • Theorem \ref{thm:cwrv}
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Lemma 2.7
  • ...and 27 more