Abstract: Let  G be a simple undirected graph with order n≥1.Denote by ρ(G) and μ(G) the spectral radius of the adjacency matrix and the Laplacian matrix of G,respectively.In this paper,by the way of spanning bipartite subgraphs,it is  showed that:let G be a simple graph with chromatic number k≥1.If k is even,ρ(G)≤(k-1)/kμ(G);if  k is odd,ρ(G)≤k/(k+1)μ(G).

Key words: spectral radius, Laplacian spectral radius, chromatic number, spanning bipartite subgraph