ELA, Volume 16, pp. 90-98, March 2007, abstract.

Interlacing for Weighted Graphs Using the Normalized 
Laplacian

Steve Butler

The problem of relating the eigenvalues of the normalized
Laplacian for a weighted graph G and G-H, for H a subgraph 
of G is considered.  It is shown that these eigenvalues 
interlace and that the tightness of the interlacing is 
dependent on the number of nonisolated vertices of H. 
Weak coverings of a weighted graph are also defined and 
interlacing results for the normalized Laplacian for such a 
covering are given. In addition there is a discussion about 
interlacing for the Laplacian of directed graphs.