Behaviors of pairs of dimensions and depths of edge ideals

TL;DR

This paper investigates the relationship between dimensions and depths of edge ideals of graphs, providing classifications for connected graphs with small vertices and chordal graphs, enhancing understanding of their algebraic properties.

Contribution

It characterizes possible dimension-depth pairs of edge ideals for connected graphs, especially for small and chordal graphs, expanding algebraic graph theory knowledge.

Findings

Complete classification for small connected graphs

Characterization of pairs for connected chordal graphs

Insights into algebraic properties of edge ideals

Abstract

Edge ideals of finite simple graphs $G$ on $n$ vertices are the ideals $I(G)$ of the polynomial ring $S$ in $n$ variables generated by the quadratic monomials associated with the edges of $G$. In this paper, we consider the possible pairs of dimensions and depths of $S/I(G)$ for connected graphs with a fixed number of vertices. We discuss such pairs in the case where dimension is relatively large. As a corollary, we completely determine the pairs for connected graphs with small number of vertices. We also study the possible pairs for connected chordal graphs.