Graphs with Minimum Algebraic Connectivity I: Proofs of Aldous-Fill and Guiduli-Mohar Conjectures

Abstract

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.