Initially Regular Sequences on Cycles and Depth of Unicyclic Graphs

TL;DR

This paper constructs initially regular sequences on specific cycles to compute their depth and extends the analysis to unicyclic graphs, providing new methods for understanding their algebraic properties.

Contribution

It introduces effective initially regular sequences for cycles of the form C_{3n+2} and computes the depth of certain unicyclic graphs, advancing algebraic graph theory.

Findings

Initially regular sequences for cycles C_{3n+2} are established.

The depth of certain unicyclic graphs is accurately computed.

A detailed analysis of associated primes of initial ideals is provided.

Abstract

In this article, we establish initially regular sequences on cycles of the form $C_{3n+2}$ for $n\ge 1$, in the sense of \cite{FHM-ini}. These sequences accurately compute the depth of these cycles, completing the case of finding effective initially regular sequences on cycles. Our approach involves a careful analysis of associated primes of initial ideals of the form $\rm{ini}_>(I,f)$ for arbitrary monomial ideals $I$ and $f$ linear sums. We describe the minimal associated primes of these ideals in terms of the minimal primes of $I$. Moreover, we obtain a description of the embedded associated primes of arbitrary monomial ideals. Finally, we accurately compute the depth of certain types of unicyclic graphs.