The distance coloring of graphs

Let G be a connected graph with maximum degree Δ ≥ 3. We investigate the upper bound for the chromatic number χγ of the power graph Gγ. It was proved that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\chi _\gamma (G) \leqslant \Delta \tfrac{{(\Delta - 1)^\gamma - 1}} {{\Delta - 2}} + 1 = :M + 1 $\end{document}, 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) ≤ 2γ − 1, (3) g(G) ≥ 2γ + 2, and the connectivity κ(G) ≥ 3 if γ ≥ 3, κ(G) ≥ 4 but g(G) > 6 if γ = 2, (4) Δ is sufficiently larger than a given number only depending on γ, then χγ(G) ≤ M − 1. By means of the spectral radius λ1(G) of the adjacency matrix of G, it was shown that χ2(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 χγ(G) < λ1(G)γ + 1 γ ≥ 3.