No abstract

Voronoi defines a partition of the cone of positive semidefinite n-ary forms2 , 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 2L 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 in R 5 given by Engel [8], we give a complete list of seven fivedimensional rigid lattices.

A subset R of a vector space V (or R n ) 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 called maximal 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 Seymour [7] and 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 A n , which are graphic, cographic systems related to non-planar 3-connected cubic graphs without separating cuts of cardinality 3, and a special system E 5 representing the matroid R 10 from [7] which 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.

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.

The hypermetric cone Hn is the cone in the space R n(n-1)/2 of all vectors d = (dij) 1