代数连通度最小的图 I：Aldous-Fill 与 Guiduli-Mohar 猜想的证明

中文摘要 AI 辅助

Aldous 和 Fill (2002) 猜想：在具有 $n$ 个顶点的连通正则图上，随机游走的松弛时间的最大值以 $(1+o(1))\frac{3n^2}{2\pi^2}$ 为上界，且当 $n$ 为偶数时达到渐近相等。由于 $d$-正则图 $G$ 的松弛时间等于 $d/\mu(G)$，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 $\delta=d\ge 3$ 以及所有足够大的阶数，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。对于固定的奇数度 $d\ge 3$，Abdi 和 Ghorbani (2004) 猜想：具有最小代数连通度的 $d$-正则图具有相同的结构。我们证明了 Aldous--Fill 猜想和 Guiduli--Mohar 猜想，以及关于固定奇数度 $d$-正则图的相应猜想。最后，我们证明：对于每个固定的奇数度 $d\ge 3$，$d$-正则图以及具有固定最小度 $d\ge 3$ 的图，若其代数连通度渐近最小，则其直径渐近最大。这确立了 Abdi 和 Ghorbani 另一个猜想的相应情形。

英文摘要

Aldous and Fill (2002) conjectured that the maximum relaxation time of a random walk on a connected regular graph with $n$ vertices is bounded above by $(1+o(1))\frac{3n^2}{2π^2}$, with asymptotic equality for even $n$. 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 $δ=d\ge 3$ 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. For fixed odd degree $d\ge 3$, Abdi and Ghorbani (2004) conjectured that $d$-regular graphs with minimum algebraic connectivity have the same structure. We prove the Aldous--Fill conjecture and the Guiduli--Mohar conjecture, as well as the corresponding conjecture for $d$-regular graphs of fixed odd degree. Finally, we prove that, for every fixed odd degree $d\ge 3$, $d$-regular graphs, as well as graphs of fixed minimum degree $d\ge 3$, whose algebraic connectivity is asymptotically minimum have asymptotically maximum diameter. This establishes the corresponding cases of another conjecture of Abdi and Ghorbani.

发表机构

• School of Mathematics, Institute for Research in Fundamental Sciences (IPM)（基础科学研究院数学学院）
• Hamburg University of Technology（汉堡工业大学）

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