Viatcheslav P. Grishukhin

Clin d'oeil on L 1 -embeddable planar graphs

In this note we present some properties of LI-embeddable planar graphs. We present a characterization of graphs isometrically embeddable into half-cubes. This result implies that every planar Li-graph G has a scale 2 embedding into a hypercube. Further, under some additional conditions we show that for a simple circuit C of a planar Li-graph G the subgraph H of G bounded by C is also Li-embeddable. In many important cases, the length of C is the dimension of the smallest cube in which H has a scale embedding. Using these facts we establish the LI-embeddability of a list of planar graphs.

THE HYPERMETRIC CONE IS POLYHEDRAL

The hypermetric coneHn is the cone in the spaceRn(n−1)/2 of all vectorsd=(dij)1≤i<j≤n satisfying the hypermetric inequalities: −1≤i≤j≤nzjzjdij≤ 0 for all integer vectorsz inZn with −1≤i≤nzi=1. We explore connections of the hypermetric cone with quadratic forms and the geometry of numbers (empty spheres andL-polytopes in lattices). As an application, we show that the hypermetric coneHn is polyhedral.

Computing extreme rays of the metric cone for seven points

Abstract We give a complete description of the extreme rays of the comesMk of all metrics onk points fork ≤ 7. The coneM7 has more than 60 000 rays, and we use a computer program to make the computation. We give four collections of facets such that it is possible to compute all extreme rays ofM7 contained in facets of these collections. Then we prove that any ray ofM7 is permutationally equivalent to one of the computed rays.

Fullerenes and coordination polyhedra versus half-cube embeddings

A fullerene Fn is a 3-regular (or cubic) polyhedral carbon molecule for which the n vertices - the carbons atoms - are arranged in 12 pentagons and (n/2 - 10) hexagons. Only a finite number of fullerenes are expected to be, up to scale, isometrically embeddable into a hypercube. Looking for the list of such fullerenes, we first check the embeddability of all fullerenes Fn for n < 60 and of all preferable fullerenes Cn for n < 86 and their duals. Then, we consider some infinite families, including fullerenes with icosahedral symmetry, which describe virus capsids, onion-like metallic clusters and geodesic domes. Quasi-embeddings and fullerene analogues are considered. We also present some results on chemically relevant polyhedra such as coordination polyhedra and cluster polyhedra. Finally we conjecture that the list of known embeddable fullerenes is complete and present its relevance to the Katsura model for vesicles cells.

Non-rigidity Degree of a Lattice and Rigid Lattices

Voronoi defines a partition of the cone of positive semidefinite n -ary formsPn into L -type domains. Each L -type domain is an open polyhedral cone of dimensionk , 1 ?k?n(n+ 1)2, where n is the number of variables and dimension of the corresponding lattice. We define a non-rigidity degree of a lattice as the dimension of the L -type domain containing the lattice. We prove that the non-rigidity degree of a lattice equals the corank of a system of equalities connecting norms of minimal vectors of cosets of 2 L in L. A lattice of non-rigidity degree 1 is called rigid. A lattice is rigid if any of its sufficiently small deformations distinct from a homothety changes its L -type. Using the list of 84 zone-contracted Voronoi polytopes inR5given by Engel 8, we give a complete list of seven five-dimensional rigid lattices.

Maximal Unimodular Systems of Vectors

A subset R of a vector space V(orRn) is called unimodular(or U -system) if every vector r?R has an integral representation in every basis B?R. A U -system R is calledmaximal if one cannot add a non-zero vector not colinear to vectors of R such that the new system is unimodular and spans RR. In this work, we refine assertions of Seymour7and give a description of maximal U -systems. We show that a maximal U -system can be obtained as amalgams (as 1- and 2-sums) of simplest maximal U -systems called components. A component is a maximal U -system having no 1- and 2-decompositions. It is shown that there are three types of components: the root systems An, which are graphic, cographic systems related to non-planar 3-connected cubic graphs without separating cuts of cardinality 3, and a special system E5representing the matroid R10from7which is neither graphic nor cographic. We give conditions that are necessary and sufficient for maximality of an amalgamated U -system. We give a complete description of all 11 maximal U -systems of dimension 6.

Allowed Boundary Sequences for Fused Polycyclic Patches and Related Algorithmic Problems

We consider sequences that encode boundary circuits of fused polycycles made up of polygonal faces with p sides, p < or = 6. We give a constructive algorithm for recognizing such sequences when p = 5 or 6. A simpler algorithm is given for planar hexagonal sequences. Hexagonal and pentagonal sequences of length at most 8 are tabulated, the former corresponding to planar benzenoid hydrocarbons CxHy with y up to 14.

There are Exactly 222 L-types of Primitive Five-dimensional Lattices

Enumerations of combinatorial types (i.e., L-types) of five-dimensional primitive lattices was performed in 6 five-dimensional primitive lattices waand 3. But both results differ in one L-type. We show explicitly the L-type missed in 6.

Properties of parallelotopes equivalent to Voronoi's conjecture

A parallelotope is a polytope whose translation copies fill space without gaps and intersections by interior points. Voronoi conjectured that each parallelotope is an affine image of the Dirichlet domain of a lattice, that is to say a Voronoi polytope. We give several properties of a parallelotope and prove that each of them is equivalent to it being an affine image of a Voronoi polytope.

How to compute the rank of a Delaunay polytope

Roughly speaking, the rank of a Delaunay polytope is its number of degrees of freedom. In [M. Deza, M. Laurent, Geometry of Cuts and Metrics, Springer Verlag, Berlin, Heidelberg, 1997], a method for computing the rank of a Delaunay polytope P, using the hypermetrics related to P, is given. Here a simpler more efficient method, which uses affine dependencies instead of hypermetrics, is given. This method is applied to the classical Delaunay polytopes: cross-polytopes and half-cubes. Then, we give an example of a Delaunay polytope, which does not have any affine basis.