最小代数连通度与最大直径：Aldous--Fill猜想与Guiduli--Mohar猜想
Minimum algebraic connectivity and maximum diameter: Aldous--Fill and Guiduli--Mohar conjectures
AI总结：
该研究探讨图的代数连通度与直径的关系，验证相关猜想对Aldous--Fill猜想的蕴含关系，证明一般形式下Guiduli问题的否定答案，并提出渐近情形下的相关猜想。
摘要：
Aldous与Fill（2002年）提出猜想：在具有$n$个顶点的连通正则图上，随机游走的最大弛豫时间为$(1+o(1)) \frac{3n^{2}}{2π^{2}}$。Guiduli与Mohar（1996年）的猜想则预测了在所有最小度$δ$为给定值$d$的连通图中，代数连通度$μ$最小的图的结构。我们证明，对于奇数$d$，该猜想蕴含Aldous--Fill猜想。我们针对具有最小$μ$的$d$-正则图的结构提出了另一个猜想，并证明对于偶数$d$，该猜想同样蕴含Aldous--Fill猜想。\n现有文献中已通过经验观察到，$μ$较小的图往往具有较大的直径。就此，Guiduli（1996年）提出问题：具有最大直径的立方图的代数连通度是否小于所有其他立方图？受这些问题启发，我们研究了具有最大直径的图与具有最小代数连通度的图之间的相互作用。我们证明，Guiduli问题的一般形式（即对每个$d\ge 3$的$d$-正则图而言）的答案是否定的。\n我们旨在建立该问题的渐近表述。研究证明，对于$d\ge 5$的$d$-正则图，以及$d\ge 4$的最小度$δ=d$的图，具有渐近最大直径的图并不一定具有渐近最小的$μ$。我们猜想，具有渐近最小$μ$的$d$-正则图（或最小度$δ=d$的图）应当具有渐近最大的直径。上述结果很大程度上依赖于我们对近最大直径图的结构的理解，以及对其代数连通度的最优估计，针对这类图的Aldous--Fill猜想也由此得证。
英文摘要：
Aldous and Fill (2002) conjectured that the maximum relaxation time for the random walk on a connected regular graph with $n$ vertices is $(1+o(1)) \frac{3n^{2}}{2π^{2}}$. A conjecture by Guiduli and Mohar (1996) predicts the structure of graphs whose algebraic connectivity $μ$ is the smallest among all connected graphs whose minimum degree $δ$ is a given $d$. We prove that this conjecture implies the Aldous--Fill conjecture for odd $d$. We pose another conjecture on the structure of $d$-regular graphs with minimum $μ$, and show that this also implies the Aldous--Fill conjecture for even $d$. In the literature, it has been noted empirically that graphs with small $μ$ tend to have a large diameter. In this regard, Guiduli (1996) asked if the cubic graphs with maximum diameter have algebraic connectivity smaller than all others. Motivated by these, we investigate the interplay between the graphs with maximum diameter and those with minimum algebraic connectivity. We show that the answer to Guiduli problem in its general form, that is for $d$-regular graphs for every $d\ge 3$ is negative. We aim to develop an asymptotic formulation of the problem. It is proven that $d$-regular graphs for $d\ge 5$ as well as graphs with $δ=d$ for $d\ge 4$ with asymptotically maximum diameter, do not necessarily exhibit the asymptotically smallest $μ$. We conjecture that $d$-regular graphs (or graphs with $δ=d$) that have asymptotically smallest $μ$, should have asymptotically maximum diameter. The above results rely heavily on our understanding of the structure as well as optimal estimation of the algebraic connectivity of nearly maximum-diameter graphs, from which the Aldous--Fill conjecture for this family of graphs also follows.
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