Laplacian eigenvalues of the second power of a graph

The k th power of a graph G , denoted by G k , is the graph with the same vertex set as G , such that two vertices are adjacent in G k if and only if their distance is at most k in G . 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 G 2 .