The main purpose of this book is to introduce algebraic, combinatorial, and computational
methods to study monomial algebras and their presentation ideals, including Stanley …

Let C be a clutter and let IC/ R be its edge ideal. This is a survey paper on the algebraic and
combinatorial properties of R= IC/and C, respectively. We give a criterion to estimate the …

In this paper, we give new upper bounds on the regularity of edge ideals whose resolutions
are k-step linear; surprisingly, the bounds are logarithmic in the number of variables. We …

A very well–covered graph is an unmixed graph without isolated vertices such that the
height of its edge ideal is half of the number of vertices. We study these graphs by means of …

We show that the co-chordal cover number of a graph 𝐺 gives an upper bound for the
Castelnuovo-Mumford regularity of the associated edge ideal. Several known combinatorial …

In this paper, we prove the existence of a special order on the set of minimal monomial
generators of powers of edge ideals of arbitrary graphs. Using this order, we find new upper …

Let G be a finite simple graph and I (G) denote the corresponding edge ideal. For all s ≥ 1
s≥ 1, we obtain upper bounds for reg (I (G)^ s) reg (I (G) s) for bipartite graphs. We then …

We survey recent studies on the Castelnuovo–Mumford regularity of edge ideals of graphs
and their powers. Our focus is on bounds and exact values of reg I (G) reg I (G) and the …

We study the relationship between depth and regularity of a homogeneous ideal I and those
of (I, f) and I: f, where f is a linear form or a monomial. Our results have several interesting …

Let G be a bipartite graph with edge ideal I (G) whose quotient ring R/I (G) is sequentially
Cohen–Macaulay. We prove:(1) the independence complex of G must be vertex …