发表机构
School of Mathematics, Institute for Research in Fundamental Sciences (IPM); Hamburg University of Technology（基础科学研究院数学学院; 汉堡工业大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文解决了偶数度正则图最小代数连通度的结构猜想，给出了精确渐近公式和直径界，并证明了渐近最小代数连通度图具有渐近最大直径。
AI 中文摘要
Aldous 和 Fill（2002）猜想连通正则图上随机游走的渐近最大松弛时间。由于 $d$-正则图 $G$ 的松弛时间为 $d/\mu(G)$，其中 $\mu(G)$ 表示其代数连通度，该猜想与在正则图中最小化代数连通度的问题密切相关。Guiduli 和 Mohar（1996）猜想，对于每个固定的最小度 $\delta=d\ge3$ 和所有足够大的阶数，具有最小代数连通度的图是路径状的，并且除了靠近其两端的有限部分外，具有规定的块结构。Abdi 和 Ghorbani（2024）对具有最小代数连通度和固定度 $d\ge3$ 的 $d$-正则图提出了类似的结构猜想。在第一部分中，我们证明了 Aldous--Fill 猜想、Guiduli--Mohar 猜想，以及对于奇数度，Abdi--Ghorbani 猜想。在本文中，我们解决了剩余的偶数度情形，从而完成了具有最小代数连通度的正则图的结构刻画。我们还证明了，对于每个固定的偶数 $d\ge4$，阶数为 $n$ 时的最小代数连通度为 $2(d-2)\pi^2/n^2+O_d(n^{-3})$，并且每个最小化图的直径为 $3n/(d+1)+O_d(1)$。对于每个固定的偶数 $d\ge6$，代数连通度渐近最小的 $d$-正则图具有渐近最大的直径。最后，我们获得了所有偶数正则度（包括随 $n$ 增长的度）的尖锐归一化间隙界。
英文摘要
Aldous and Fill (2002) conjectured the asymptotic maximum relaxation time of a random walk on a connected regular graph. Since the relaxation time of a $d$-regular graph $G$ is $d/μ(G)$, where $μ(G)$ denotes its algebraic connectivity, this conjecture is closely related to the problem of minimizing algebraic connectivity among regular graphs. Guiduli and Mohar (1996) conjectured that, for every fixed minimum degree $δ=d\ge3$ and all sufficiently large orders, graphs with minimum algebraic connectivity are path-like and, apart from bounded portions near their two ends, have a prescribed block structure. Abdi and Ghorbani (2024) proposed an analogous structural conjecture for $d$-regular graphs with minimum algebraic connectivity and fixed degree $d\ge3$. In Part~I, we proved the Aldous--Fill conjecture, the Guiduli--Mohar conjecture, and for odd degrees, the Abdi Ghorbani conjecture. In this paper, we settle the remaining even-degree case, thereby completing the structural characterization of regular graphs with minimum algebraic connectivity. We also prove that, for every fixed even $d\ge4$, the minimum algebraic connectivity at order $n$ is $2(d-2)π^2/n^2+O_d(n^{-3})$, and that every minimizing graph has diameter $3n/(d+1)+O_d(1)$. For every fixed even $d\ge6$, $d$-regular graphs whose algebraic connectivity is asymptotically minimum have asymptotically maximum diameter. Finally, we obtain a sharp normalized-gap bound for all even regular degrees, including degrees that grow with $n$.
