AI 中文总结
本文完全解决了Kolokolnikov提出的关于$n$个顶点、$2n-4$条边的图的代数连通性上界及极大化图的猜想。
AI 中文摘要
对于图$G$，设$\u03b1(G)$为$G$的拉普拉斯矩阵的第二小特征值，也称为代数连通性，它在刻画图的连通性和网络的收敛性中发挥重要作用。Kolokolnikov猜想：在所有有$n$个顶点、恰好$2n-4$条边的图中，$\u03b1(G)\u22642$，且一个极大化图是两部分大小分别为2和$n-2$的完全二部图。本文完全解决了该猜想。
英文摘要
For a graph $G$, let $α(G)$ be the second smallest eigenvalue of the Laplacian matrix of $G$, also known as the algebraic connectivity. Algebraic connectivity plays an important role in characterizing the connectivity of graphs and convergence properties of networks. Kolokolnikov conjectured that among all graphs on $n$ vertices with exactly $2n-4$ edges, $α(G)\leq 2$ and one of the maximizers is the complete bipartite graph whose two parts have sizes two and $n-2$, respectively. In this paper, we completely resolve this conjecture.
