Monomial ideals of graphs with loops

TL;DR

This paper studies monomial edge ideals of graphs with loops, identifying classes with linear resolutions, analyzing their algebraic invariants, and exploring the structure and properties of their vertex cover ideals.

Contribution

It introduces new classes of graphs with monomial edge ideals that have linear resolutions and characterizes their vertex cover ideals, including cases with loops on vertices.

Findings

Identified classes of graphs with monomial ideals having linear resolutions

Computed algebraic invariants for these ideals

Proved vertex cover ideals are of linear type

Abstract

We investigate, using the notion of linear quotients, significative classes of connected graphs whose monomial edge ideals, not necessarily squarefree, have linear resolution, in order to compute standard algebraic invariants of the polynomial ring related to these graphs modulo such ideals. Moreover we are able to determine the structure of the ideals of vertex covers for the edge ideals associated to the previous classes of graphs which can have loops on any vertex. Lastly, it is showed that these ideals are of linear type.