Is Locality preserving projections (LPP) method the same as Laplacian eigenmap method?
There is a subtle difference, which can be difficult to recognize. They both minimize the same objective: 

$\sum_{i,j} w_{ij} || y_i - y_j ||^2_2$

However, they parametrize the predictor on y differently. Locality preserving projections is linear because the parameterization is linear: $y_i = x_iW$. Laplacian eigenmaps is non-linear because the parametrization uses a kernel: $y_i = k_iA$. Kernel LPP is equivalent to Laplacian eigenmaps.  

For a good description and a summary showing the connections between all these different dimensionality reduction algorithms, see the paper:

TRACE OPTIMIZATION AND EIGENPROBLEMS IN DIMENSION REDUCTION METHODS, KOKIOPOULOU et al.

or the summary table in Section 2.7 in the thesis:

Regularized Factor Models, Martha White