AI 中文总结
本文针对图论领域，证明了Kolokolnikov于2015年提出的关于特定阶数和边数的图的最大代数连通性的猜想。
AI 中文摘要
图G的代数连通性是一种被广泛研究的图不变量，与图的连通性、扩张性等其他性质相关。给定n和m，α(n,m)是具有n个顶点、m条边的图的最大代数连通性。2015年，Kolokolnikov猜想当n≥3时，α(n,2n−4)=2，且通过计算验证了n≤12时该结论成立，本文证明了Kolokolnikov的这一猜想。
英文摘要
The algebraic connectivity of a graph $G$ is a well-studied graph invariant that is related to other properties of the graph such as connectivity and expansion. Given $n$ and $m$, $α(n,m)$ is the maximum algebraic connectivity of a graph with $n$ vertices and $m$ edges. In 2015, Kolokolnikov conjectured that $α(n,2n-4)=2$ for $n\geq 4$, and verified this claim computationally for $n \le 12$. In this paper, we prove Kolokolnikov's conjecture. We also show that $α(n,3(n-3)) = 3$ is false in general. %Combined with the computational verification for $n \le 12$, this yields $α(n,2n-4)=2$ for all admissible values of $n$.
CommentsCorrected minor typos and included the appendix
