It is known that there exists a Minimum Distance Diagram (MDD) for circulant digraphs of degree two (or double-loop computer networks) which is an L-shape. Its description provides the diameter and the average distance of the graph on constant time. In this paper we clarify, justify and extend these diagrams to circulant digraphs of arbitrary degree by presenting monomial ideals as a natural to...

We provide some new conditions under which the graded Betti numbers of a mono-mial ideal can be computed in terms of the graded Betti numbers of smaller ideals, thus complementing Eliahou and Kervaire's splitting approach. As applications, we show that edge ideals of graphs are splittable, and we provide an iterative method for computing the Betti numbers of the cover ideals of Cohen-Macaulay b...

Backelin proved that the multigraded Poincaré series for resolving a residue field over a polynomial ring modulo a monomial ideal is a rational function. The numerator is simple, but until the recent work of Berglund there was no combinatorial formula for the denominator. Berglund’s formula gives the denominator in terms of ranks of reduced homology groups of lower intervals in a certain lattic...

We propose several new distances for phylogenetic trees based on two new algebraic representations of them as permutations and as monomial ideals.

We survey the Stanley-Reisner correspondence in combinatorial commutative algebra, describing fundamental applications involving Alexander duality, associated primes, f and h-vectors, and Betti numbers of monomial ideals.

The purpose of this paper is to present a family of CohenMacaulay monomial ideals such that their integral closures have embedded components and hence are not Cohen-Macaulay.

We show that the mixed volumes of arbitrary convex bodies are equal to multiplicities graded families monomial ideals, and normalized limits ideals. This result evinces close relation between theories from geometry commutative algebra.

Let R = k[x1, ..., xn] be the polynomial ring in n independent variables, where k is a field. In this work we will study Bass numbers of local cohomology modules H I (R) supported on a squarefree monomial ideal I ⊆ R. Among them we are mainly interested in Lyubeznik numbers. We build a dictionary between the modules H I (R) and the minimal free resolution of the Alexander dual ideal I∨ that all...

In this paper we give new upper bounds on the regularity of edge ideals whose resolutions are k-steps linear; surprisingly, the bounds are logarithmic in the number of variables. We also give various bounds for the projective dimension of such ideals, generalizing other recent results. By Alexander duality, our results also apply to unmixed square-free monomial ideals of codimension two. We als...

We prove the multiplicity bounds conjectured by Herzog-HunekeSrinivasan and Herzog-Srinivasan in the following cases: the strong conjecture for edge ideals of forests, and the weaker Taylor bound conjecture for all quadratic monomial ideals. We determine when equality holds in the conjectured bound, and verify that when equality holds, the resolution is pure. We characterize forests that have C...