C. Marijuán

Minimal systems of generators for ideals of semigroups

We give an algorithmic method to compute a minimal system of generators for I, in the general case of a subsemigroup S of a finitely generated abelian group, such that S∩(—S) = 0.

Minimal strong digraphs

We introduce adequate concepts of expansion of a digraph to obtain a sequential construction of minimal strong digraphs. We characterize the necessary and sufficient condition for an external expansion of a minimal strong digraph to be a minimal strong digraph. We prove that every minimal strong digraph of order n>=2 is the expansion of a minimal strong digraph of order n-1 and we give sequentially generative procedures for the constructive characterization of the classes of minimal strong digraphs. Finally we describe algorithms to compute unlabeled minimal strong digraphs and their isospectral classes.

FINITE TOPOLOGIES AND DIGRAPHS

In this paper we study the relation between finite topologies and digraphs. We associate a digraph to a topology by means of the “specialization” relation between points in the topology. Reciprocally, we associate a topology to each digraph, taking the sets of vertices adjacent (in the digraph) to v, for all vertex v, as a subbasis of closed sets for the topology. We use these associations to examine the relation between a simple digraph and its homologous topology. We also extend this relation to the functions preserving the structure between these classes of objects.

Monomial Inequalities for Newton Coefficients and Determinantal Inequalities for p-Newton Matrices

We consider Newton matrices for which the Newton coefficients are positive. We show that one monomial in these coefficients dominates another for all such Newton matrices if and only if a certain generalized form of majorization occurs. As the Newton coefficients may be viewed as average values of principal minors of a given size, these monomial inequalities may be interpreted as determinantal inequalities in such familiar classes as the positive definite, totally positive, and M-matrices, etc.

Symmetric nonnegative realizability via partitioned majorization

A sufficient condition for symmetric nonnegative realizability of a spectrum is given in terms of (weak) majorization of a partition of the negative eigenvalues by a selection of the positive eigenvalues. If there are more than two positive eigenvalues, an additional condition, besides majorization, is needed on the partition. This generalizes observations of Suleǐmanova and Loewy about the cases of one and two positive eigenvalues, respectively. It may be used to provide insight into realizability of 5-element spectra and beyond.

Birational equivalence of reduced graphs

Abstract We study a canonical desingularization process for oriented reduced graphs. We use it to give an arithmetical characterization for these graphs by means of sequences of natural numbers, based on the representation of a partial ordering by its maximal chains. For such graphs we define, in analogy with Algebraic Geometry, similar tools and language as in birational geometry. We study a class of birationally equivalent graphs giving it a graph structure and describing, in an explicit way, a canonical graph in the class with a minimal number of points.

Blowing Up Acyclic Graphs and Geometrical Configurations

Blowing up is a useful technique in algebraic and analytic geometry. In particular, it is the main tool for proving resolution of singularities. Hironaka [2] proved in 1964 that every algebraic variety over a field of characteristic zero admits a resolution of singularities which is obtained by successive blowing ups of certain regular centers. Moreover, he proves the stronger version of embedded resolution of singularities, i.e., for every (singular) subvariety X of a smooth variety Z there exists a sequence of birational morphisms

Comparison on the spectral radii of weighted digraphs that differ in a certain subdigraph

{Z_N} \to {Z_{N - 1}} \to \cdots \to {Z_1} \to {Z_0} = Z,