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Classes of EDMs

  • Abdo Y. Alfakih
Chapter

Abstract

Euclidean Distance Matrices fall into two classes: spherical and nonspherical. The first part of this chapter discusses various characterizations and several subclasses of spherical EDMs. Among the examples of spherical EDMs discussed are: regular EDMs, cell matrices, Manhattan distance matrices, Hamming distance matrices on the hypercube, distance matrices of trees and resistance distance matrices of electrical networks. The second part focuses on nonspherical EDMs and their characterization. As an interesting example of nonspherical EDMs, we discuss multispherical EDMs.

References

  1. 12.
    A.Y. Alfakih, A remark on the Manhattan distance matrix of a rectangular grid, 2012. arXiv/1208.5150Google Scholar
  2. 18.
    A.Y. Alfakih, H. Wolkowicz, Two theorems on Euclidean distance matrices and Gale transform. Linear Algebra Appl. 340, 149–154 (2002)MathSciNetCrossRefGoogle Scholar
  3. 32.
    R. Balaji, R.B. Bapat, On Euclidean distance matrices. Linear Algebra Appl. 424, 108–117 (2007)MathSciNetCrossRefGoogle Scholar
  4. 33.
    R.B. Bapat, Graphs and Matrices (Springer, London, 2010)CrossRefGoogle Scholar
  5. 73.
    P.G. Doyle, J.L. Snell, Random Walks and Electric Networks (Mathematical Association of America, Washington, 1984)Google Scholar
  6. 79.
    M. Fiedler, A geometric approach to the Laplacian matrix of a graph, in Combinatorial and Graph-Theoretical Problems in Linear Algebra, ed. by R.A. Brualdi, S. Friedland, V. Klee (Springer, New York, 1993), pp. 73–98CrossRefGoogle Scholar
  7. 81.
    M. Fiedler, Moore-Penrose involutions in the classes of Laplacians and simplices. Linear Multilinear Algebra 39, 171–178 (1995)MathSciNetCrossRefGoogle Scholar
  8. 92.
    J.C. Gower, Euclidean distance geometry. Math. Sci. 7, 1–14 (1982)MathSciNetzbMATHGoogle Scholar
  9. 93.
    J.C. Gower, Properties of Euclidean and non-Euclidean distance matrices. Linear Algebra Appl. 67, 81–97 (1985)MathSciNetCrossRefGoogle Scholar
  10. 94.
    R.L. Graham, L. Lovász, Distance matrix polynomials of trees. Adv. Math. 29, 60–88 (1978)MathSciNetCrossRefGoogle Scholar
  11. 95.
    R.L. Graham, H.O. Pollak, On the addressing problem for loop switching. Bell Syst. Tech. J. 50, 2495–2519 (1971)MathSciNetCrossRefGoogle Scholar
  12. 96.
    R.L. Graham, P.M. Winkler, On isometric embeddings of graphs. Trans. Am. Math. Soc. 288, 527–536 (1985)MathSciNetCrossRefGoogle Scholar
  13. 101.
    T.L. Hayden, P. Tarazaga, Distance matrices and regular figures. Linear Algebra Appl. 195, 9–16 (1993)MathSciNetCrossRefGoogle Scholar
  14. 104.
    T.L. Hayden, J. Lee, J. Wells, P. Tarazaga, Block matrices and multispherical structure of distance matrices. Linear Algebra Appl. 247, 203–216 (1996)MathSciNetCrossRefGoogle Scholar
  15. 110.
    A.J. Hoffman, On the polynomial of a graph. Am. Math. Mon. 70, 30–36 (1963)CrossRefGoogle Scholar
  16. 116.
    G. Jaklič, J. Modic, On properties of cell matrices. Appl. Math. Comput. 216, 2016–2023 (2010)MathSciNetzbMATHGoogle Scholar
  17. 119.
    H.W.E. Jung, Ueber die kleinste kugel, die eine raumliche figur einschliesst. J. Reine Angew. Math. 123, 241–257 (1901)MathSciNetzbMATHGoogle Scholar
  18. 121.
    D.J. Klein, M. Randić, Resistance distance. J. Math. Chem. 12, 81–95 (1993)MathSciNetCrossRefGoogle Scholar
  19. 123.
    H. Kurata, S. Matsuura, Characterization of multispherical and block structures of Euclidean distance matrices. Linear Algebra Appl. 439, 3177–3183 (2013)MathSciNetCrossRefGoogle Scholar
  20. 124.
    H. Kurata, T. Sakuma, A group majorization ordering for Euclidean distance matrices. Linear Algebra Appl. 420, 586–595 (2007)MathSciNetCrossRefGoogle Scholar
  21. 125.
    H. Kurata, P. Tarazaga, Multispherical Euclidean distance matrices. Linear Algebra Appl. 433, 534–546 (2010)MathSciNetCrossRefGoogle Scholar
  22. 146.
    H. Mittelmann, J. Peng, Estimating bounds for quadratic assignment problems associated with Hamming and Manhattan distance matrices based on semidefinite programming. SIAM J. Optim. 20, 3408–3426 (2010)MathSciNetCrossRefGoogle Scholar
  23. 152.
    A. Neumaier, Distances, graphs and designs. Eur. J. Combin. 1, 163–174 (1980)MathSciNetCrossRefGoogle Scholar
  24. 153.
    A. Neumaier, Distance matrices, dimension and conference graphs. Nederl. Akad. Wetensch. Indag. Math. 43, 385–391 (1981)MathSciNetCrossRefGoogle Scholar
  25. 175.
    S. Seshu, M.B. Reed, Linear Graphs and Electrical Networks (Addison-Wesley, Reading, 1961)zbMATHGoogle Scholar
  26. 182.
    G.P.H. Styan, G.E. Subak-Sharpe, Inequalities and equalities associated with the Campbell-Youla generalized inverse of the indefinite admittance matrix of resistive networks. Linear Algebra Appl. 250, 349–370 (1997)MathSciNetCrossRefGoogle Scholar
  27. 184.
    P. Tarazaga, Faces of the cone of Euclidean distance matrices: characterizations, structure and induced geometry. Linear Algebra Appl. 408, 1–13 (2005)MathSciNetCrossRefGoogle Scholar
  28. 186.
    P. Tarazaga, T.L. Hayden, J. Wells, Circum-Euclidean distance matrices and faces. Linear Algebra Appl. 232, 77–96 (1996)MathSciNetCrossRefGoogle Scholar
  29. 188.
    P. Tarazaga, B. Sterba-Boatwright, K. Wijewardena, Euclidean distance matrices: special subsets, systems of coordinates and multibalanced matrices. Comput. Appl. Math. 26, 415–438 (2007)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  • Abdo Y. Alfakih
    • 1
  1. 1.Department of Mathematics and StatisticsUniversity of WindsorWindsorCanada

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