连通图的无符号拉普拉斯谱半径的界  

The Bound for the Signless Laplacian Spectral Radius of Connected Graph

作　　者：杨小波 王国平[1] YANG Xiao-bo;WANG Guo-ping(School of Mathematical Sciences,Xinjiang Normal University,Urumqi,Xinjiang,830017,China)

机构地区：[1]新疆师范大学数学科学学院,新疆乌鲁木齐830017

出　　处：《新疆师范大学学报（自然科学版）》2025年第2期61-66,共6页Journal of Xinjiang Normal University(Natural Sciences Edition)

摘　　要：假设G=(V,E)是n阶连通图,λ_(1)(G)表示G的无符号拉普拉斯谱半径。对于任意点v_(i)∈V(G),其在G中的邻点的集合用N_(i)表示,并且点v_(i)在G中的邻点的数目用d_(i)表示,G的最大度用Δ(G)表示。文章展示了当连通图G的最大度为Δ(G)<n-1时,则λ_(1)(G)≥max{m_(i)'+(1+(m_(i)'-1)^(2)/d_(2,i))d_(i)/m_(i)':v_(i)∈V(G)},其中m_(i)'=Σv_(i)v_(j)∈E(G)(d_(j)-|N_(i)⋂N_(j)|)/d_(i);令d_(2,i)表示G中与点v_(i)距离为2的点的数目,则λ_(1)(G)≥max{(p_(ij)+(1-p_(ij))^(2)/p_(ij)max{1,d_(i)-1}|N_(i)⋃N_(j)|:v_(i)v_(j)∈E(G),d_(i)≥d_(j)},其中p_(ij)=e(N_(i),N_(j)-N_(i))/d_(i)(|N_(i)⋃N_(j)|-d_(i));如果G是简单连通图,则λ_(1)(G)≤√2Δ(G)(Δ(G)-1)+1+1.Let G=(V,E)be the simple connected graph of order n andλ_(1)(G)be the spectral radius of signless Laplacian of G.For each v_(i)∈V(G),the set of its neighbors in G is denoted by N_(i),and the number of neighbors of v_(i)in G is denoted by d_(i).The maximum degree of G is defined byΔ(G).In this paper,it is showm that if G is connected graph andΔ(G)<n-1,thenλ_(1)(G)≥max{m_(i)'+(1+(m_(i)'-1)^(2)/d_(2,i))d_(i)/m_(i)':v_(i)∈V(G)}where m_(i)'=Σv_(i)v_(j)∈E(G)(d_(j)-|N_(i)⋂N_(j)|)/d_(i)and d_(2,i)is the number of vertices at distance two from v_(i)in G.Also it is shown thatλ_(1)(G)≥max{(p_(ij)+(1-p_(ij))^(2)/p_(ij)max{1,d_(i)-1}|N_(i)⋃N_(j)|:v_(i)v_(j)∈E(G),d_(i)≥d_(j)},Where p_(ij)=e(N_(i),N_(j)-N_(i))/d_(i)(|N_(i)⋃N_(j)|-d_(i)).Then it is also shown that if G is a connected graph,thenλ_(1)(G)≤√2Δ(G)(Δ(G)-1)+1+1.

关 键 词：无符号拉普拉斯矩阵 谱半径 连通图 

分 类 号：O157.5[理学—数学]