INTRODUCTION An n-point metric space (X; D) can be represented by an n n table specifying the distances. Such tables arise in many diverse areas. For example, consider the following scenario in microbiology: X is a collection of bacterial strains, and for every two strains, one is given their dissimilarity (computed, say, by comparing their DNA). It is dicult to see any structure in a large table of numbers, and so we would like to represent a given metric space in a more comprehensible way. For example, it would be very nice if we could assign to each x 2 X a point f(x) in the plane in such a way that D(x; y) equals the Euclidean distance of f(x) and f(y). Such a representation would allow us to see the structure of the metric space: tight clusters, isolated points, and so on. Another advantage would be that the metric would now be represented by only 2n real numbers, the coordinates of the n points in the plane, instead of numbers as before. Moreover, many quantities concern

In the last decade many important applications of eigenvalues and eigenvectors of graphs in combinatorial optimization were discovered. The number and importance of these results is so fascinating that it makes sense to present this survey.

We group in this paper, within a unified framework, many applications of the following polyhedra: cut, boolean quadric, hypermetric and metric polyhedra. We treat, in particular, the following applications: ffl ` 1 - and L 1 -metrics in functional analysis, ffl the max-cut problem, the Boole problem and multicommodity flow problems in combinatorial optimization, ffl lattice holes in geometry of numbers, ffl density matrices of many-fermions systems in quantum mechanics. We present some other applications, in probability theory, statistical data analysis and design theory.

. A finite semimetric d on a set X is hypermetric if it satisfies the inequality P i;j2X b i b j d ij 0 for all b 2 Z X with P i2X b i = 1. Hypermetricity turns out to be the appropriate notion for describing the metric structure of holes in lattices. We survey hypermetrics, their connections with lattices and applications. 2 M. Deza, V.P. Grishukhin and M. Laurent Contents 1 Introduction 2 Preliminaries 2.1 Distance spaces Metric notions Operations on distance spaces Preliminary results on distance spaces 2.2 Lattices and L-polytopes Lattices L-polytopes L-polytopes and Voronoi polytopes Lattices and positive quadratic forms L-polytopes and empty ellipsoids Basic facts on L-polytopes Construction of L-polytopes L-polytopes in dimension k 4 2.3 Finiteness of the number of types of L-polytopes in given dimension 3 Hypermetrics and L-polytopes 3.1 The connection between hypermetrics and L-polytopes 3.2 Polyhedrality of the hypermetric cone 3.3 L-polytopes in root lattic...

A hierarchy of classes of graphs is proposed which includes hypercubes, acyclic

Let e be an edge of a median graph G which is contained in exactly one 4-cycle.

This paper is concerned with the problem of locating n new facilities in the median space when there are k facilities already located. The objective is to minimize the weighted sum of distances. Necessary and sufficient conditions are established. Based on these results a polynomial algorithm is presented. The algorithm requires the solution of a sequence of minimum--cut problems. The complexity of this algorithm for median graphs and networks and for finite median spaces with jV j points is O(jV j 3 + jV j/(n)), where /(n) is the complexity of the applied maximum-flow algorithm. For a simple rectilinear polygon P with N edges and equipped with the rectilinear distance the analogical algorithm requires O(N+k(logN+logk+/(n))) time and O(N+k/(n)) time in the case of the vertex-restricted multifacility location problem. 1.

is the graph with vertices representing #-classes of G, two classes being adjacent if they cross on some cycle in G. The following problem posed in [11, Problem 7.1] is considered: what can be said about the partial cube G if G is the join A of not edge-less graphs A and B? It is proved that for arbitrary graphs A and B, where at least one of them contains an = A#B. On the other hand, if G is a median graph, then G B if and only if G = H #K, = B. Along the way some new facts about partial cubes are obtained.

We show how to test whether a graph with n vertices and m edges is a partial cube, and if so how to find a distance-preserving embedding of the graph into a hypercube, in the near-optimal time bound O(n²), improving previous O(nm)-time solutions.

Afinite metric is h − embeddable if it can be embedded isometrically in the N-cube (hypercube) for some N. Itisknown that the problem of testing whether ametric is h − embeddable is NP-Complete, even ifthe distances are restricted to the set {2, 4, 6}. Here we study the problem where the distances are restricted to the set {1, 2, 3} and give a polynomial time algorithm and forbidden submetric characterisation. In fact, we show these metrics are h − embeddable if and only if they are 11 − gonal and the sum of the distances arround any triangle is even. The so-called truncated metric case, where the distances are chosen from {1, 2} is particularly simple, the only embeddable metrics arise from the graphs K 1,n−1, K 2,2, and 2K n ( K n with all distances 2). 1.