Depth of edge ideals and vertex connectivity of finite graphs

TL;DR

This paper establishes sharp lower bounds for the depth of edge ideals of graph complements and their powers, relating algebraic properties to graph connectivity and size.

Contribution

It provides new precise lower bounds for the depth of edge ideals and their powers based on the graph's vertex connectivity and size.

Findings

Sharp lower bounds for depth of $S/I(G^c)$ in terms of $n$ and $\, ext{connectivity}$

Sharp lower bounds for depth of $S/I(G^c)^2$ and $S/I(G^c)^{(2)}$

Results connect algebraic invariants with combinatorial graph properties.

Abstract

Let $G$ be a finite graph on $[n]:=\{1, \ldots, n\}$ and $\kappa(G)$ its vertex connectivity. Let $S=K[x_1, \ldots, x_n]$ denote the polynomial ring in $n$ variables over a field $K$ and $I(G^c)$ the edge ideal of the complementary graph $G^c$ of $G$. It is a classical result that ${\rm depth} S/I(G^c) \leq \kappa(G) + 1$. We give a sharp lower bound of ${\rm depth} S/I(G^c)$ in terms of $n$ and $\kappa(G)$. Furthermore, a sharp lower bound of ${\rm depth} S/I(G^c)^2$ as well as that of ${\rm depth} S/I(G^c)^{(2)}$ in terms of $n$ and $\kappa(G)$ is given.