HOMOLOGICAL LINEAR QUOTIENTS AND EDGE IDEALS OF GRAPHS

It is well known that the edge ideal I(G) of a simple graph G has linear quotients if and only if G (c) is chordal. We investigate when the property of having linear quotients is inherited by homological shift ideals of an edge ideal. We will see that adding a cluster to the graph G (c) when I(G)) has homological linear quotients results in a graph with the same property. In particular, I(G) has homological linear quotients when G (c) is a block graph. We also show that adding pinnacles to trees preserves the property of having homological linear quotients for the edge ideal of their complements. Furthermore, I(G) has homological linear quotients for every graph G such that G (c) is a lambda-minimal chordal graph.