Binomial Edge Ideals of Unicyclic Graphs

TL;DR

This paper investigates the algebraic properties of binomial edge ideals of unicyclic graphs, establishing bounds and characterizations for depth and regularity, and analyzing extremal Betti numbers for various graph structures.

Contribution

It provides new bounds and characterizations for the depth and regularity of binomial edge ideals in unicyclic graphs, including extremal Betti number analysis.

Findings

Depth of $S/J_G$ is at least $n$ for unicyclic graphs.

Characterization of graphs with specific depth values.

Bounds on regularity for graphs formed by attaching whiskers.

Abstract

Let $G$ be a connected simple graph on the vertex set $[n]$. Banerjee-Betancourt proved that $depth(S/J_G)\leq n+1$. In this article, we prove that if $G$ is a unicyclic graph, then the depth of $S/J_G$ is bounded below by $n$. Also, we characterize $G$ with $depth(S/J_G)=n$ and $depth(S/J_G)=n+1$. We then compute one of the distinguished extremal Betti numbers of $S/J_G$. If $G$ is obtained by attaching whiskers at some vertices of the cycle of length $k$, then we show that $k-1\leq reg(S/J_G)\leq k+1$. Furthermore, we characterize $G$ with $reg(S/J_G)=k-1$, $reg(S/J_G)=k$ and $reg(S/J_G)=k+1$. In each of these cases, we classify the uniqueness of extremal Betti number of these graphs.