Advertisement

Some Recent Results on Signed Graphs with Least Eigenvalues ≥ -2

  • G. R. Vijayakumar
  • N. M. Singhi
Part of the The IMA Volumes in Mathematics and Its Applications book series (IMA, volume 20)

Abstract

A survey of some results concerning the class of sigraphs represented by root-systems D n, n ∈ N and E 8 is given and some unsolved problems are described.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [CDS]
    D. Cvetkovic, M. Doob and S. Simic, Generalized line graphs, Journal of Graph Theory, 5 (1981), pp. 385–399.MathSciNetzbMATHCrossRefGoogle Scholar
  2. [CV]
    P.D. Chawathe and G.R. Vijayakumar, Signed graphs represented by D , submitted.Google Scholar
  3. [CGSS]
    P.J. Cameron, J.M. Goethals, J.J. Seidel and E.E. Shult, Line graphs, root systems and elliptic geometry, J. Alg., 43 (1976), pp. 305–327.MathSciNetzbMATHCrossRefGoogle Scholar
  4. [H1]
    F. Harary, “Graph Theory”, Addison-Wesley, Reading Mass, 1972.Google Scholar
  5. [H2]
    A.J. Hoffman, -1 - √2? in “Combinatorial Structures and their Applications”, R. Guy. Ed. Gordon and Breech, New York (1970) 173–176.Google Scholar
  6. [H3]
    A.J. Hoffman, On graphs whose least eigenvalue exceeds -1 - √2, J. Linear Algebra and its Applications, 16 (1977).Google Scholar
  7. [HR]
    A.J. Hoffman and D.K. Ray Chaudhuri, On a spectral characterization of regular line graphs, unpublished, manuscript (1965).Google Scholar
  8. [RSV]
    S.B. Rao, N.M. Singhi and K.S. Vijayan, the minimal forbidden graphs for generalized line graphs, “Proceedings of International Symposium in Combinatorics, Calcutta 1980, Ed. S.B. Rao, Springer Verlag lecture notes No. 885, 459–472.Google Scholar
  9. [RW]
    A. Van Rooij and H. Wilf, The interchange graph of a finite graph, Acta. Math. Acad. Scie. Hungar., 16 (1965), pp. 263–269.zbMATHCrossRefGoogle Scholar
  10. [S]
    J.J. Seidel, Strongly regular graphs with (-1,1,0)-adjacency matrix having eigenvalue 3, Linear Algebra Appl. 1 (1968), pp. 281–298.MathSciNetzbMATHCrossRefGoogle Scholar
  11. [V]
    G.R. Vijayakumar, Signed graphs represented by D, Europ. J. Comb, 8 (1987), pp. 103–112.MathSciNetzbMATHGoogle Scholar
  12. [VRS]
    Vijayakumar, S.B. Rao and N.M. Singhi, Graphs with eigenvalues at least -2, Linear Algebra and its Applications, 46 (1982), pp. 27–42.Google Scholar
  13. [W]
    E. Witt, Spiegelungsgruppan and Auf zahlung halbein facher Leicher Ringe, Abh. Math. Sem. Hamburg, 14 (1941), pp. 289–337.MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag New York, Inc. 1990

Authors and Affiliations

  • G. R. Vijayakumar
    • 1
  • N. M. Singhi
    • 1
  1. 1.School of MathematicsTata Institute of Fundamental ResearchColaba, BombayIndia

Personalised recommendations