Effect of different graph operations at algebraic connectivity of graph laplacian?
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.