Let $D$ be a weighted oriented graph, whose underlying graph is $G$, and let $I(D)$ be its edge ideal. If $G$ has no $3$-, $5$-, or $7$-cycles, or $G$ is K{o}nig, we characterize when $I(D)$ is unmixed. If $G$ has no $3$- or $5$-cycles, or $G$ is Kon
ig, we characterize when $I(D)$ is Cohen--Macaulay. We prove that $I(D)$ is unmixed if and only if $I(D)$ is Cohen--Macaulay when $G$ has girth greater than $7$ or $G$ is Konig and has no $4$-cycles.
Following Britz, Johnsen, Mayhew and Shiromoto, we consider demi-ma-troids as a(nother) natural generalization of matroids. As they have shown, demi-ma-troids are the appropriate combinatorial objects for studying Weis duality. Our results here appor
t further evidence about the trueness of that observation. We define the Hamming polynomial of a demimatroid $M$, denoted by $W(x,y,t)$, as a generalization of the extended Hamming weight enumerator of a matroid. The polynomial $W(x,y,t)$ is a specialization of the Tutte polynomial of $M$, and actually is equivalent to it. Guided by work of Johnsen, Roksvold and Verdure for matroids, we prove that Betti numbers of a demimatroid and its elongations determine the Hamming polynomial. Our results may be applied to simplicial complexes since in a canonical way they can be viewed as demimatroids. Furthermore, following work of Brylawski and Gordon, we show how demimatroids may be generalized one step further, to combinatroids. A combinatroid, or Brylawski structure, is an integer valued function $rho$, defined over the power set of a finite ground set, satisfying the only condition $rho(emptyset)=0$. Even in this extreme generality, we will show that many concepts and invariants in coding theory can be carried on directly to combinatroids, say, Tutte polynomial, characteristic polynomial, MacWilliams identity, extended Hamming polynomial, and the $r$-th generalized Hamming polynomial; this last one, at least conjecturelly, guided by the work of Jurrius and Pellikaan for linear codes. All this largely extends the notions of deletion, contraction, duality and codes to non-matroidal structures.
Let $G$ be a connected graph and let $mathbb{X}$ be the set of projective points defined by the column vectors of the incidence matrix of $G$ over a field $K$ of any characteristic. We determine the generalized Hamming weights of the Reed--Muller-typ
e code over the set $mathbb{X}$ in terms of graph theoretic invariants. As an application to coding theory we show that if $G$ is non-bipartite and $K$ is a finite field of ${rm char}(K) eq 2$, then the $r$-th generalized Hamming weight of the linear code generated by the rows of the incidence matrix of $G$ is the $r$-th weak edge biparticity of $G$. If ${rm char}(K)=2$ or $G$ is bipartite, we prove that the $r$-th generalized Hamming weight of that code is the $r$-th edge connectivity of $G$.
We study the complete intersection property and the algebraic invariants (index of regularity, degree) of vanishing ideals on degenerate tori over finite fields. We establish a correspondence between vanishing ideals and toric ideals associated to nu
merical semigroups. This correspondence is shown to preserve the complete intersection property, and allows us to use some available algorithms to determine whether a given vanishing ideal is a complete intersection. We give formulae for the degree, and for the index of regularity of a complete intersection in terms of the Frobenius number and the generators of a numerical semigroup.
We show that the degree of a graded lattice ideal of dimension 1 is the order of the torsion subgroup of the quotient group of the lattice. This gives an efficient method to compute the degree of this type of lattice ideals.
For the family of graded lattice ideals of dimension 1, we establish a complete intersection criterion in algebraic and geometric terms. In positive characteristic, it is shown that all ideals of this family are binomial set theoretic complete inters
ections. In characteristic zero, we show that an arbitrary lattice ideal which is a binomial set theoretic complete intersection is a complete intersection.
We compute the basic parameters (dimension, length, minimum distance) of affine evaluation codes defined on a cartesian product of finite sets. Given a sequence of positive integers, we construct an evaluation code, over a degenerate torus, with pres
cribed parameters. As an application of our results, we recover the formulas for the minimum distance of various families of evaluation codes.
Let C be a clutter and let A be its incidence matrix. If the linear system x>=0;xA<=1 has the integer rounding property, we give a description of the canonical module and the a-invariant of certain normal subrings associated to C. If the clutter is a
connected graph, we describe when the aforementioned linear system has the integer rounding property in combinatorial and algebraic terms using graph theory and the theory of Rees algebras. As a consequence we show that the extended Rees algebra of the edge ideal of a bipartite graph is Gorenstein if and only if the graph is unmixed.
Let C be a uniform clutter, i.e., all the edges of C have the same size, and let A be the incidence matrix of C. We denote the column vectors of A by v1,...,vq. The vertex covering number of C, denoted by g, is the smallest number of vertices in any
minimal vertex cover of C. Under certain conditions we prove that C is vertex critical. If C satisfies the max-flow min-cut property, we prove that A diagonalizes over the integers to an identity matrix and that v1,...,vq is a Hilbert basis. It is shown that if C has a perfect matching such that C has the packing property and g=2, then A diagonalizes over the integers to an identity matrix. If A is a balanced matrix we prove that any regular triangulation of the cone generated by v1,...,vq is unimodular. Some examples are presented to show that our results only hold for uniform clutters. These results are closely related to certain algebraic properties, such as the normality or torsion freeness, of blowup algebras of edge ideals and to finitely generated abelian groups. They are also related to the theory of Grobner bases of toric ideals and to Ehrhart rings.
Let G be a simple graph and let J be its ideal of vertex covers. We give a graph theoretical description of the irreducible b-vertex covers of G, i.e., we describe the minimal generators of the symbolic Rees algebra of J. Then we study the irreducibl
e b-vertex covers of the blocker of G, i.e., we study the minimal generators of the symbolic Rees algebra of the edge ideal of G. We give a graph theoretical description of the irreducible binary b-vertex covers of the blocker of G. It is shown that they correspond to irreducible induced subgraphs of G. As a byproduct we obtain a method, using Hilbert bases, to obtain all irreducible induced subgraphs of G. In particular we obtain all odd holes and antiholes. We study irreducible graphs and give a method to construct irreducible b-vertex covers of the blocker of G with high degree relative to the number of vertices of G.
