Let G be a simple connected graph, and k be a positive integer. The kth graph power \(G^k\) of G is the graph whose vertex set is the vertex set of G and two distinct vertices are adjacent if and only if their distance in G is at most k. The independence number of \(G^k\) is the maximum size of an independent set in \(G^k\) . The chromatic number of \(G^k\) , \(\chi (G^k)\) , is the smallest number of colors needed to color \(G^k\) so that the vertices sharing the same color form an independent set of \(G^k\) . In this paper, we study the independence number of \(G^k\) , when G is a tree, by giving a known lower bound for the number of vertices that a tree must have, assuming a certain independence number of \(G^k\) . As a consequence of our proof, we describe all trees that attain this lower bound. For some of the trees given G, we also describe a ( \(\chi (G^k)\) )-coloring of \(G^k\) .
