Abstract: Let G be a simple graph with n vertices, the new upper bound on the sum of the Q spectral radius of a graph and its complement were given by its m edges, minimal degree of a vertex δ, maximal degree of a vertex Δ and chromatic number k． When graph G has no isolated vertex ,we have 2(n-1)≤ρ(Q(G))+ρ(Q([AKG-]))≤2(Δ-δ+n-1) and ρ(Q(G))+ρQ([AKG-]))≤2n-3+[KF(][JB((]2-[SX(]1[]2[SX)][JB))](n-1)n[KF)]，where t=min{k,[AKk-]}. When graph [AKG-] has l isolated vertices ,we have ρ(Q(G))+ρ(Q([AKG-]))≤2n-3+[KF(][JB((]2-[SX(]1[]k[SX)][JB))](n-1)2+l[KF)]. At the same time the upper bound on the sum of the Laplace spectral radius of a graph and its complement was given.

Key words: Graph, Complement graph, Chromatic number, spectral radius, Laplace spectral radius, Upper bound

摘要： 设G为n阶简单图，利用边数m，最小、最大顶点度δ和Δ以及色数k给出了G与其补图[AKG-]的Q谱半径之和的上界，当G不含孤立点时有：2(n-1)≤ρ(Q(G))+ρ(Q([AKG-]))≤2(Δ-δ+n-1)和ρ(Q(G))+ρQ([AKG-]))≤2n-3+[KF(][JB((]2-[SX(]1[]2[SX)][JB))](n-1)n[KF)]，其中t=min{k,[AKk-]}。当[AKG-]含l个孤立点时有：ρ(Q(G))+ρ(Q([AKG-]))≤2n-3+[KF(][JB((]2-[SX(]1[]k[SX)][JB))](n-1)2+l[KF)]，同时给出了图G与其补图[AKG-]的拉普拉斯谱半径之和的一个上界。

关键词: 图, 补图, 色数, Q谱半径, 拉普拉斯谱半径, 上界