关于图的代数连通度的注记

摘要

Let G be a connected graph of order n whose algebraic connectivity, vertex connectivity, and edge connectivity are α(G), κ(G), and λ(G), respectively. First, an equivalent condition is given for α(G)=κ(G) when 2κ(G)n-2. And the algebraic multiplicity of α(G) and the property of eigenvectors corresponding to α(G) are discussed when the above equality holds. By the obtained results, the equivalent condition for α(G)=λ(G) is also established when 1λ(G)n-2.%n阶连通图G的代数连通度、点连通度和边连通度分别记作α(G), κ(G)和λ(G).本文给出了当2≤κ(G)≤n-2时,α(G)=κ(G)成立的充要条件,讨论了α( G) 的代数重数以及相应于特征值α(G)的特征向量的性质.最后给出了当 1≤λ(G)≤n-2时,α(G)=λ(G)的充要条件.