On the $\mathrm{v}$-number of binomial edge ideals of some classes of graphs

TL;DR

This paper investigates the $ ext{v}$-number of binomial edge ideals for various classes of graphs, providing explicit computations, bounds, characterizations, and confirming a conjecture for powers of such ideals with linear resolutions.

Contribution

It computes the $ ext{v}$-number for Cohen-Macaulay closed graphs, provides bounds for cycles and trees, characterizes graphs with $ ext{v}$-number 2, and proves a conjecture on $ ext{v}$-numbers of powers of binomial edge ideals.

Findings

Computed $ ext{v}$-number for Cohen-Macaulay closed graphs.

Established bounds for cycles and binary trees.

Characterized graphs with $ ext{v}$-number 2.

Abstract

Let $G$ be a finite simple graph, and $J_G$ denote the binomial edge ideal of $G$. In this article, we first compute the $\mathrm{v}$-number of binomial edge ideals corresponding to Cohen-Macaulay closed graphs. As a consequence, we obtain the $\mathrm{v}$-number for paths. For cycle and binary tree graphs, we obtain a sharp upper bound for $\mathrm{v}(J_G)$ using the number of vertices of the graph. We characterize all connected graphs $G$ with $\mathrm{v}(J_G) = 2$. We show that for a given pair $(k,m), k\leq m$, there exists a graph $G$ with an associated monomial edge ideal $I$ having $\mathrm{v}$-number equal to $k$ and regularity $m$. If $2k \leq m$, then there exists a binomial edge ideal with $\mathrm{v}$-number $k$ and regularity $m$. Finally, we compute $\mathrm{v}$-number of powers of binomial edge ideals with linear resolution, thus proving a conjecture on the $\mathrm{v}$-number of powers of a graded ideal having linear powers, for the class of binomial edge ideals.