Regularity of powers of d-sequence (parity) binomial edge ideals of unicycle graphs

TL;DR

This paper classifies unicycle graphs based on the $d$-sequence property of their binomial edge ideals and studies the regularity of their powers, providing insights into algebraic properties of these graph ideals.

Contribution

It offers a complete classification of unicycle graphs with $d$-sequence and parity $d$-sequence binomial edge ideals, and analyzes the regularity of their powers.

Findings

Classified all unicycle graphs with $d$-sequence binomial edge ideals.

Classified unicycle graphs with parity $d$-sequence binomial edge ideals.

Studied the regularity behavior of powers of these ideals.

Abstract

We classify all unicycle graphs whose edge-binomials form a $d$-sequence, particularly linear type binomial edge ideals. We also classify unicycle graphs whose parity edge-binomials form a $d$-sequence. We study the regularity of powers of (parity) binomial edge ideals of unicycle graphs generated by $d$-sequence (parity) edge-binomials.