Let G be a connected graph with maximum degree △≥ 3. We investigate the upper bound for the chromatic number Xr（G） of the power graph Gr. It was proved that Xr（G） ≤ △ （△-1）r-1/△-2 ＋1 =M ＋ 1, where the equality holds if and only if G is a Moore graph. If G is not a Moore graph, and G satisfies one of the following conditions： （1） G is non-regular, （2） the girth g（G）≤2r - 1, （3） g（G） ≥ 2r＋ 2, and the connectivity k（G） ≥ 3 if r ≥ 3, k（G）≥4 but g（G） 〉 6 if r= 2, （4） A is sufficiently larger than a given number only depending on % then Xr（G） ≤ M - 1. By means of the spectral radius λ1（G） of the adjacency matrix of G, it was shown that X2（G） ≤λ1（G）2＋ 1, where the equality holds if and only if G is a star or a Moore graph with diameter 2 and girth 5, and Xr（G） 〈 λ1（G）r＋ 1 if ≥ 3.
