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连通图的无符号拉普拉斯谱半径的界

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假设G=(V,E)是n阶连通图,λ1(G)表示G的无符号拉普拉斯谱半径.对于任意点vi∈V(G),其在G中的邻点的集合用Ni表示,并且点vi在G中的邻点的数目用di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λ1(G)≥max{mi'+(1+(mi'-1)2/d2,i)di/mi':vi∈V(G)},其中mi'=∑vivj∈E(G)(dj-|Ni∩Nj|)/di;令d2,i表示G中与点vi距离为2的点的数目,则λ1(G)≥max{(pij+(1-pij)2/pij max{1,di-1})|Ni∪Nj|:vivj∈E(G),di≥dj},其中pij=e(Ni,Nj-Ni)/di(|Ni∪Nj|-di);如果G是简单连通图,则λ1(G)≤√2Δ(G)(Δ(G)-1)+1+1.
The Bound for the Signless Laplacian Spectral Radius of Connected Graph
Let G=(V,E) be the simple connected graph of order n and λ1(G) be the spectral radius of signless Laplacian of G. For each vi ∈ V(G),the set of its neighbors in G is denoted by Ni,and the number of neighbors of vi in G is denoted by di. Thte maximum degree of G is definedd by Δ(G). In this paper,it is showm that if G is connected graph and Δ(G)<n-1,then λ1(G)≥max{mi'+(1+(mi'-1)2/d2,i)di/mi':vi∈V(G) },where mi'=∑vivj∈E(G)(dj-|Ni∩Nj|)/di and d2,i is the number of vertices at distance two from vi in G. Also it is shown that λ1(G)≥max{(pij+(1-pij)2/pij max{1,di-1})|Ni∪Nj|:vivj∈E(G),di ≥ dj},Where pij=e(Ni,Nj-Ni)/di(|Ni∪Nj|-di). Then it is also shown that if G is a connected graph,then λ1(G)≤√2Δ(G)(Δ(G)-1)+1+1.

Signless Laplacian MatrixSpectral radiusConnected graph

杨小波、王国平

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新疆师范大学 数学科学学院,新疆 乌鲁木齐 830017

无符号拉普拉斯矩阵 谱半径 连通图

2025

新疆师范大学学报(自然科学版)
新疆师大学报

新疆师范大学学报(自然科学版)

影响因子:0.457
ISSN:1008-9659
年,卷(期):2025.44(2)