The $v$-number and Castelnuovo-Mumford regularity of graphs

TL;DR

This paper establishes a family of connected graphs where the difference between the $v$-number and the Castelnuovo-Mumford regularity can be arbitrarily large, demonstrating a specific relationship between these graph invariants.

Contribution

It constructs explicit examples of connected graphs with a prescribed difference between the $v$-number and regularity, showing the unbounded nature of this difference.

Findings

For each integer $k\, extgreater\,0$, a connected graph $H_k$ exists with $v(H_k)=reg(H_k)+k$.

The difference between $v$-number and regularity can be arbitrarily large.

The relationship between $v$-number and regularity is not bounded by a fixed constant.

Abstract

We prove that for every integer $k\geq 1$, there exists a connected graph $H_k$ such that $v(H_k)=reg(H_k)+k$, where $v(G)$ and $reg(G)$ denote the $v$-number and the (Castelnuovo-Mumford) regularity of a graph $G$ respectively.