Effect of different graph operations at algebraic connectivity of graph laplacian? The algebraic connectivity of a graph G is the second-smallest eigenvalue of the Laplacian matrix of G. This eigenvalue is greater than 0 if and only if G is a connected graph. The magnitude of this value reflects how well connected the overall graph is.

for an example, "adding self-loops" does not change laplacian eigenvalues (specially  algebraic connectivity) of graph.
Because, laplacian(G)= D-A is invariant with respect to adding self-loops.

My question is:

Does anyone has studied effect of different operations (such as edge contraction) on spectrum of laplacian?
do you know good references?

Remark: the exact definition of the algebraic connectivity depends on the type of Laplacian used.  For this question I prefer to use Fan Chung definition in SPECTRAL GRAPH THEORY. In this book Fan Chung has uesed a rescaled version of the Laplacian, eliminating the dependence on the number of vertices.
Intuitively operations that preserve connectivity will not decrease the eigenvalues. For example, adding edges to the graph does not decrease the connectivity. 

In general, if H is a subgraph of a graph G, by interlacing we know that the i-th largest Laplacian eigenvalue of H is no larger than the i-th largest Laplacian eigenvalue of G. A proof can be found in Proposition 3.2.1 of the book "Spectra of graphs" by Brouwer and Haemers. Note that the definition of Laplacian used in the book is not normalized; it has node degrees on the diagonal and -1 (or 0 if there is no edge) in the off-diagonal entries.