On the Laplacian coefficients of graphs under some transformations

We generalize three approaches on graph transformations, respectively, from Stevanović and Ilić (2009) [16] and Tan (2011) [19]. We also generalize an approach of graph transformations on the spectral radius of adjacency matrix into the Laplacian coefficients of graphs from Li and Feng (1979) [26]. Moreover, we determine the unique tree having the third maximal Laplacian coefficients among all n-vertex trees.