Minimum algebraic connectivity and maximum diameter: Aldous--Fill and Guiduli--Mohar conjectures

Aldous and Fill (2002) conjectured that the maximum relaxation time for the random walk on a connected regular graph with $n$ vertices is $(1+o(1)) \frac{3n^{2}}{2\pi ^{2}}$. A conjecture by Guiduli and Mohar (1996) predicts the structure of graphs whose algebraic connectivity $\mu$ is the smallest among all connected graphs whose minimum degree $\delta$ 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 $\mu$, 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 $\mu$ 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 $\delta =d$ for $d\ge 4$ with asymptotically maximum diameter, do not necessarily exhibit the asymptotically smallest $\mu$. We conjecture that $d$-regular graphs (or graphs with $\delta =d$) that have asymptotically smallest $\mu$, 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.