Torsten Sillke, 1990
A Conjecture of Nordhaus type:

  Let G be a graph with n vertices.
  Let m = m(G) be the number of eigenvalues
  of the adjacency matrix of G which are lower or equal -1.
  Let v = v(G) be the number of eigenvalues
  of the adjacency matrix of G which are lower 0.
  Let m' = m(G') and v' = v(G') with G' the complement of G.


  1/2 n  <=  m + m'  <=  n-1  <=  v + v'  <=  3/2 n - 2


References:

- Cvetkovic, D., Doob, and Sachs;
  Spectra of Graphs
  Academic Press 1978

- Cvetkovic, D., Rowlinson, P., and Simic, S.;
  Eigenspaces of graphs. 
  Cambridge U. Press.  1997.  ISBN 0 521 57352 1. 

- E. A. Nordhaus, J. W. Gaddum;
  On Complementary Graphs,
  Amer. Math. Monthly 63 (1956) 175-177