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Derived eigenvalues of symmetric matrices, with applications to distance geometry. (English) Zbl 0706.15009
Let \(A\in R^{n\times n}\) and j be the all-one vector of dimension n. The derived eigenvalues of the symmetric matrix A are the zeros of \(P_ A(s)=-j^ Tadj (Is-A)j\). With this notion a new characterization for the embeddability of a finite metric space into Euclidean space is given. In more detail the particular case of two-distance sets is discussed.
Reviewer: M.Voicu

MSC:
15A18 Eigenvalues, singular values, and eigenvectors
51K05 General theory of distance geometry
15B57 Hermitian, skew-Hermitian, and related matrices
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[1] Blokhuis, A., Few-distance sets, (1984), Reidel Dordrecht · Zbl 0548.51014
[2] Cvetković, D.M.; Doob, M.; Sachs, H., Spectra of graphs, (1980), Academic New York · Zbl 0458.05042
[3] Haemers, W.H., Eigenvalue techniques in design and graph theory, (1980), Reidel Dordrecht · Zbl 0429.05013
[4] Hoffman, A.J., Eigenvalues of graphs, (), 225-245
[5] Larman, D.G.; Rogers, A.A.; Seidel, J.J., On 2-distance sets in Euclidean space, Bull. London math. soc., 9, 261-267, (1977) · Zbl 0399.51011
[6] Menger, U., Untersuchungen über allgemeine metrik, Math. ann., 100, 75-163, (1928) · JFM 54.0622.02
[7] Neumaier, A., Distances, graphs and designs, European J. combin., 1, 163-174, (1980) · Zbl 0435.05015
[8] Neumaier, A., Distance matrices, dimension, and conference graphs, Proc. kon. nederl. akad. wetensch. ser. A, 84, 385-391, (1981), ( = Indag. Math. 43) · Zbl 0523.05019
[9] Neumaier, A., Graph representations, two-distance sets, and equiangular lines, Linear algebra appl., 114/115, 141-156, (1989) · Zbl 0724.05043
[10] A. Neumaier and F. Bussemaker, Exceptional graphs with smallest eigenvalue -2 and related problems, to appear. · Zbl 0770.05060
[11] Parlett, B.N., The symmetric eigenvalue problem, (1980), Prentice-Hall Englewood Cliffs, N.J · Zbl 0431.65016
[12] Schoenberg, I.J., Linkages and distance geometry, Indag. math., 31, 1, 43-63, (1969) · Zbl 0169.24701
[13] Seidel, J.J., Quasiregular two-distance sets, Indag. math., 31, 1, 64-70, (1969) · Zbl 0167.50801
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