Abstract
Two sets of vertices of a hypercubes in ℝn and ℝm are said to be equivalent if there exists a distance preserving linear transformation of one hypercube into the other taking one set to the other. A set of vertices of a hypercube is said to be weakly rigid if up to equivalence it is a unique realization of its distance pattern and it is called rigid if the same holds for any multiple of its distance pattern. A method of describing all rigid and weakly rigid sets of vertices of hypercube of a given size is developed. It is also shown that distance pattern of any rigid set is on the face of convex cone of all distance patterns of sets of vertices in hypercubes.
Rigid pentagons (i.e. rigid sets of size 5 in hypercubes) are described. It is shown that there are exactly seven distinct types of rigid pentagons and one type of rigid quadrangle. It is also shown that there is a unique weakly rigid pentagon which is not rigid. An application to the study of all rigid pentagons and quadrangles inL 1 having integral distance pattern is also given.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Assouad, P. Deza, M.: Espaces metriques plongeables dans un hypercube. Ann. Discrete Math.8, 197–210 (1980)
Avis, D.: On the extreme rays of the metric cone. Canad. J. Math.32, 126–144 (1980)
Avis, D.: Hypermetric spaces and the Hamming cone. Canad. J. Math.33, 795–802 (1981)
Deza, M. (Tylkin): Realizability of matrices of distances in unitary cubes (in Russian). Probl. Kibern.7, 31–42 (1962)
Deza, M.: Matrices de formes quadratiques non-negatives pour des arguments binaries, C.R. Acad. Sci. Paris, Ser. A–B, 873–875 (1973)
Deza M.: Isometries of the hypergraphs. In: Proc. Int. Conference on Theory of Graphs, 1976 Calcutta, edited by A.R. Rao, pp. 174–189. India: McMillan 1979
Deza, M.: Small pentagonal spaces. Rend. Semin. Math. Brescia7, 269–282 (1984)
Deza, M., Frankl, P. Singhi, N.M.: On functions of strengtht. Combinatorica3, 331–339 (1983)
Deza, M., Rosenberg, I.G.: Intersection and distance patterns. Util. Math25, 191–214 (1984)
Deza, M., Singhi, N.M.: A simplicial complex associated withZ-modules and convex cones (under preparation)
Kelly, T. B.: Hypermetric spaces. Lect. Notes Math.490, 17–31 (1975)
Rosenberg, I.G.: Structural rigidity I. Ann. Discrete Math.8, 143–160 (1980)
Schoenberg, I.J.: Metric space and positive definite functions. Trans. Amer. Math. Soc.44, 522–536 (1938)
Author information
Authors and Affiliations
Additional information
This work was done during a visit of both the authors to Mehta Research Institute, Allahabad, India.
Rights and permissions
About this article
Cite this article
Deza, M., Singhi, N.M. Rigid pentagons in hypercubes. Graphs and Combinatorics 4, 31–42 (1988). https://doi.org/10.1007/BF01864151
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01864151