Eigenvalues of the k-th power of a graph

The k-th power of a graph G, denoted by G^k, is a graph with the same set of vertices as G such that two vertices are adjacent in G^k if and only if their distance in G is at most k. In this paper, we give the bounds on the spectral radius of T^k and G^k (k ≥ 1). The Nordhaus–Gaddum-type inequality for the spectral radius of the graph G^k is also presented. Moreover, we obtain an upper bound on the energy of the second power of graphs.