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