图的Laplacian谱半径界的可达性  

On Sharp Bounds for the Laplacian Spectral Radius of Graphs

作　　者：束金龙[1] 闻人凯[1] 

机构地区：[1]华东师范大学数学系,上海200062

出　　处：《华东师范大学学报（自然科学版）》2001年第3期19-24,共6页Journal of East China Normal University(Natural Science)

基　　金：国家自然科学基金资助项目 (199710 2 7) ;国家教育部骨干教师基金资助

摘　　要：设G为n阶连通的简单图 ,ρ(G)为图G的邻接谱半径 ,μ(G)表示G的Laplacian谱半径。(d1,d2 ,… ,dn) (其中d1≥d2 ≥…≥dn)为G的顶点度序列 ,令r=max{d(u) +d(v) ｜ (u ,v) ∈E(G) } =d(x) +d(y) ,s=max{d(u) +d(v)｜ (u ,v) ∈E(G) - (x ,y) }。该文证明了μ(G)上下界的可达性 :μ(G) =μ≤ 2 + ρ(LG) ,等式成立当且仅当G是偶图。μ(G)≤ 2 + (r- 2 ) (s- 2 ) ,成立等式当且仅当G为半正则偶图或P4 。μ(G)≥d1+ 1,成立等式当且仅当d1=n- 1。Let G be a connected simple graph with n vertices. Let ρ(G) and μ(G) be the adjacency spectral radius and the Laplacian spectral radius respectively. We denote the degree sequence of G by (d 1,d 2,...,d n) , where d 1≥d 2≥...≥d n . Let r= max {d(u)+d(v)|(u,v)∈E(G)}=d(x)+d(y),s= max {d(u)+d(v)|(u,v)∈E(G)-(x,y)}. In this paper, we show the sharpbounds of μ(G) . μ(G)=μ≤2+ρ(L G) , the equality holds if and if G is a bipartite graph. μ(G)≤2+(r-2)(s-2) , the equality holds if and if G is a semiregular graph or P 4 . μ(G)≥d 1+1 , the equality holds if and if d 1=n-1 .

关 键 词：邻接谱半径 LAPLACIAN谱半径 线图 半正则偶图 连通图 偶分划 可达性 

分 类 号：O157.5[理学—数学]