Laplacian eigenvalues of the second power of a graph

The kth power of a graph GG, denoted by GkGk, is the graph with the same vertex set as GG, such that two vertices are adjacent in GkGk if and only if their distance is at most kk in GG. In this paper, we give bounds on the first two largest Laplacian eigenvalues of the second power of a general graph, and on the second power of a tree. We also give a Nordhaus–Gaddum-type inequality for the Laplacian spectral radius of G2G2.