Symbolic defects of edge ideals of unicyclic graphs

TL;DR

This paper investigates the symbolic defects of edge ideals in unicyclic graphs with a unique odd cycle, providing exact calculations, classification, and Hilbert function computations for specific graph classes.

Contribution

It introduces a method to compute symbolic defects of edge ideals in unicyclic graphs and classifies graphs where certain algebraic annihilations occur.

Findings

Exact values of all symbolic defects for unicyclic graphs with odd cycles

A method to find the associated quasi-polynomial of symbolic defects

Classification of graphs where powers of the maximal ideal annihilate certain modules

Abstract

We introduce the concept of minimum edge cover for an induced subgraph in a graph. Let $G$ be a unicyclic graph with a unique odd cycle and $I=I(G)$ be its edge ideal. We compute the exact values of all symbolic defects of $I$ using the concept of minimum edge cover for an induced subgraph in a graph. We describe one method to find the quasi-polynomial associated with the symbolic defects of edge ideal $I$. We classify the class of unicyclic graphs when some power of maximal ideal annihilates $ I^{(s)}/I^s $ for any fixed $ s $. Also for those class of graphs, we compute the Hilbert function of the module $I^{(s)}/I^s$ for all $s.$