On the geodetic number of median graphs

A set of vertices S in a graph is called geodetic if every vertex of this graph lies on some shortest path between two vertices from S. In this paper, minimum geodetic sets in median graphs are studied with respect to the operation of peripheral expansion. Along the way geodetic sets of median prisms are considered and median graphs that possess a geodetic set of size two are characterized.     

A tropical toolkit

We give an introduction to Tropical Geometry and prove some results in tropical intersection theory. The first part of this paper is an introduction to tropical geometry aimed at researchers in Algebraic Geometry from the point of view of degenerations of varieties using projective not-necessarily-normal toric varieties. The second part is a foundational account of tropical intersection theory with proofs of some new theorems relating it to classical intersection theory.     

Clique partitions of distance multigraphs

We consider the minimum number of cliques needed to partition the edge set of D(G), the distance multigraph of a simple graph G. Equivalently, we seek to minimize the number of elements needed to label the vertices of a simple graph G by sets so that the distance between two vertices equals the cardinality of the intersection of their labels. We use a fractional analogue of this parameter to find lower bounds for the distance multigraphs of various classes of graphs. Some of the bounds are shown to be exact.     

Pfaffian labelings and signs of edge colorings

We relate signs of edge-colorings (as in classical Penrose’s result) with “Pfaffian labelings”, a generalization of Pfaffian orientations, whereby edges are labeled by elements of an Abelian group with an element of order two. In particular, we prove a conjecture of Goddyn that all k-edge-colorings of a k-regular Pfaffian graph G have the same sign. We characterize graphs that admit a Pfaffian labeling in terms of bricks and braces in their matching decomposition and in terms of their drawings in the projective plane. Partially supported by NSF grants 0200595 and 0354742.     

The geometry of a bi-Lagrangian manifold

This is a survey on bi-Lagrangian manifolds, which are symplectic manifolds endowed with two transversal Lagrangian foliations. We also study the non-integrable case (i.e., a symplectic manifold endowed with two transversal Lagrangian distributions). We show that many different geometric structures can be attached to these manifolds and we carefully analyze the associated connections. Moreover, we introduce the problem of the intersection of the two leaves, one of each foliation, through a point and show a lot of significative examples.     

Algebraically flat or projective algebras

We define and study algebraically flat algebras in order to have a better understanding of algebraically projective algebras of finite type (the projective algebras of literature). A close examination of the differential properties of these algebras leads to our main structure theorem. As a corollary, we get that an algebraically projective algebra of finite type over a field is either a polynomial ring or the affine algebra of a complete intersection.     

On the connectedness of the complement of the zero-divisor graph of a poset

Abstract

In this paper, connectedness is completely characterized for the complements of the zero-divisor graphs of partially ordered sets. These results are applied to annihilating ideal graphs and intersection graphs of submodules, generalizing some of the work that has recently appeared in the literature.     

The largest Erdős-Ko-Rado sets of planes in finite projective and finite classical polar spaces

Erd?s-Ko-Rado sets of planes in a projective or polar space are non-extendable sets of planes such that every two have a non-empty intersection. In this article we classify all Erd?s-Ko-Rado sets of planes that generate at least a 6-dimensional space. For general dimension (projective space) or rank (polar space) we give a classification of the ten largest types of Erd?s-Ko-Rado sets of planes. For some small cases we find a better, sometimes complete, classification.     

On linear sets on a projective line

Linear sets generalise the concept of subgeometries in a projective space. They have many applications in finite geometry. In this paper we address two problems for linear sets: the equivalence problem and the intersection problem. We consider linear sets as quotient geometries and determine the exact conditions for two linear sets to be equivalent. This is then used to determine in which cases all linear sets of rank 3 of the same size on a projective line are (projectively) equivalent. In (Donati and Durante, Des Codes Cryptogr, 46:261–267), the intersection problem for subgeometries of PG(n, q) is solved. The intersection of linear sets is much more difficult. We determine the intersection of a subline PG(1, q) with a linear set in PG(1, q ^h ) and investigate the existence of irregular sublines, contained in a linear set. We also derive an upper bound, which is sharp for odd q, on the size of the intersection of two different linear sets of rank 3 in PG(1, q ^h ).     

Harbourne constants and conic configurations on the projective plane

In this note we exhibit the so‐called Harbourne constants which capture and measure the bounded negativity on various birational models of an algebraic surface. We show an estimation for Harbourne constants for conic configurations on the complex projective plane with transversal intersection points.     

Intersections ofk-element sets

LetF be a collection ofk-element sets with the property that the intersection of no two should be included in a third. We show that such a collection of maximum size satisfies .2715k+o(k)≦≦log₂ |F|≦.7549k+o(k) settling a question raised by Erdős. The lower bound is probabilistic, the upper bound is deduced via an entropy argument. Some open questions are posed. This research has been supported in part by the Office of Naval Research under Contract N00014-76-C-0366. Supported in part by a NSF postdoctoral Fellowship.     

Trees with extremal numbers of maximal independent sets including the set of leaves

A subset S of vertices of a graph G is independent if no two vertices in S are adjacent. In this paper we study maximal (with respect to set inclusion) independent sets in trees including the set of leaves. In particular the smallest and the largest number of these sets among n-vertex trees are determined characterizing corresponding trees. We define a local augmentation of trees that preserves the number of maximal independent sets including the set of leaves.     

Computations in differential and difference modules

Constructive methods based on the Gröbner bases theory have been used many times in commutative algebra over the past 20 years, in particular, they allow the computation of such important invariants of manifolds given by systems of algebraic equations as their Hilbert polynomials. In differential and difference algebra, the analogous roles play characteristic sets.In this paper, algorithms for computations in differential and difference modules, which allow for the computation of characteristic sets (Gröbner bases) in differential, difference, and polynomial modules and differential (difference) dimension polynomials, are described. The algorithms are implemented in the algorithmic language REFAL.     

On the intersection problem for linear sets in the projective line

The aim of this paper is to investigate the intersection problem between two linear sets in the projective line over a finite field. In particular, we analyze the intersection between two clubs with possibly different maximum fields of linearity. We also consider the intersection between a certain linear set of maximum rank and any other linear set of the same rank. The strategy relies on the study of certain algebraic curves whose rational points describe the intersection of the two linear sets. Among other geometric and algebraic tools, function field theory and the Hasse–Weil bound play a crucial role. As an application, we give asymptotic results on semifields of BEL-rank two.     

Der Satz von Tits für PGL2(R), R ein kommutativer Ring vom stabilen Rang 2

Certain permutation groups on sets with distance relation are characterized as groups of projectivities PGL₂(R) on the projective line over a commutative ring R of stable rank 2, thus generalizing a classical result of Tits where R is a field.

     

Planar sections of the quadric of Lie cycles and their Euclidean interpretations

All cycles (points, oriented circles, and oriented lines of a Euclidean plane) are represented by points of a three dimensional quadric in four dimensional real projective space. The intersection of this quadric with primes and planes are, respectively, two- and one-dimensional systems of cycles. This paper is a careful examination of the interpretation, in terms of systems of cycles in the Euclidean plane, of fundamental incidence configurations involving this quadric in projective space. These interpretations yield new and striking theorems of Euclidean geometry.     

Constructions of projective linear codes by the intersection and difference of sets

Projective linear codes are a special class of linear codes whose dual codes have minimum distance at least 3. Projective linear codes with only a few weights are useful in authentication codes, secret sharing schemes, data storage systems and so on. In this paper, two constructions of q-ary linear codes are presented with defining sets given by the intersection and difference of two sets. These constructions produce several families of new projective two-weight or three-weight linear codes. As applications, our projective codes can be used to construct secret sharing schemes with interesting access structures, strongly regular graphs and association schemes with three classes.     

Properties of intersecting families of ordered sets

The first part of this paper deals with families of ordered k-tuples having a common element in different position. A quite general theorem is given and special cases are considered. The second part deals with t families of sets with some intersection properties, and generalizes results by Bollobás, Frankl, Lovász and Füredi to this case.