Abstract
A finite distance spaceX, d d: X 2 → ℤ is hypermetric (of negative type) if ∑a x a y d(x, y) ≤ 0 for all integral sequences{a x ∣x ∈ X} that sum to 1 (sum to 0).X, d is connected if the set {(x, y)∣d(x, y) = 1, x, y ∈ X} is the edge set for a connected graph onX, and graphical ifd is the path length distance for this graph. Then we prove
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Assouad, P.: Sur les inegalites valides dansL 1. Europ. J. Comb.5, 99–112 (1984)
Assouad, P.: Un espace hypermetrique non plongeable dans un espaceL 1. C. R. Acad. Sci. Paris285, 361–362 (1977)
Assouad, P., Deza, M.: Espaces metriques plonqeables dans un hypercube: aspects combinatoires. Ann. Discrete Math.8, 197–210 (1980)
Avis, D.M.: Hypermetric spaces and the Hamming cone. Canad. J. Math.33, 795–802 (1981)
Bannai, E., Ito, T.: Algebraic combinatorics 1. Association Schemes, Benjamin-Cummings Lecture Note 58. New York: Benjamin-Cummings 1984
Bourbaki, N.: Groupes et Algebres de Lie. Chap. 4–6. Paris: Hermann 1968
Bussemaker, F.C., Cvetkovic, D.M., Seidel, J.J.: Graphs related to the exceptional root systems. In: T.H. Report 76-WSK-05. Eindhoven 1976
Cameron, P.J., Goethals, J.M., Seidel, J.J., Shult, E.E.: Line graphs, root systems, and elliptic geometry. J. Algebra43, 305–327 (1976)
Coxeter, H.M.S.: Extreme forms. Canad. J. Math.3, 391–441 (1951)
Deza (Tylkin) M., On the Hamming geometry of unitary cubes. Dokl. Acad. Nauk. SSSR134, 1037–1040 (1960)
Deza, M., Singhi, N.M.: Rigid pentagons in hypercubes. Graphs and Combinatorics (to appear)
Humphreys, J.E.: Introduction to Lie Algebras and Representation Theory. New York: Springer-Verlag 1970
Kelly, J.B.: Hypermetric spaces. In: The Geometry of Metric and Linear Spaces, Lecture notes in Math. 490, pp. 17–31. Berlin: Springer-Verlag 1975
Neumaier, A.: Characterization of a class of distance-regular graphs. J. Reine Angew. Math.357, 182–192 (1985)
Schoenberg, I.J.: Metric spaces and positive definite functions. Trans. Amer. Math. Soc.44, 522–536 (1938)
Terwilliger, P.: Root systems and the Johnson and Hamming graphs. Europ. J. Comb. (to appear)
Terwilliger, P.: Root system graphs. Linear Algebra Appl. (to appear)
Winkler, P.M.: Isometric embedding in products of complete graphs. Discrete Appl. Math.7, 221–225 (1984)
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The first author was partially supported by NSF grant DMS 8600882.
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Terwilliger, P., Deza, M. The classification of finite connected hypermetric spaces. Graphs and Combinatorics 3, 293–298 (1987). https://doi.org/10.1007/BF01788552
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DOI: https://doi.org/10.1007/BF01788552