Cochordal zero divisor graphs and Betti numbers of their edge ideals

TL;DR

This paper introduces the type sequence for cochordal graphs to compute Betti numbers of their edge ideals and classifies integers with cochordal zero divisor graphs, providing explicit algebraic invariants.

Contribution

It develops a new combinatorial invariant called the type sequence for cochordal graphs and uses it to compute Betti numbers of their edge ideals, also classifying integers with cochordal zero divisor graphs.

Findings

Computed all graded Betti numbers for cochordal graph edge ideals

Classified all positive integers with cochordal zero divisor graphs

Established a link between graph properties and algebraic invariants

Abstract

We associate a sequence of positive integers, termed the type sequence, with a cochordal graph. Using this type sequence, we compute all graded Betti numbers of its edge ideal. We then classify all positive integer $n$ such that the zero divisor graph of $\mathbb{Z}/n \mathbb{Z}$ is cochordal and determine all the graded Betti numbers of its edge ideal.