Expansion properties of Whitehead moves on cubic graphs

TL;DR

We study the directed graph $\Gamma_n$ of cubic graphs on $2n$ vertices linked by Whitehead moves. The paper proves two main results: $\Gamma_n$ is connected and its outer-conductance $\phi_{out}(\Gamma_n) \to 0$ as $n\to\infty$, showing it is not an outer-expander. The connectivity is established by reducing graphs to $\,(n,b)\,-$lollipop trees and then to binary rooted trees via Whitehead-move realizable rotations, while the conductance bound uses a bridge analysis and asymptotics for bridged graphs, together with probabilistic results. Additional contributions include a Perron–Frobenius-type description of path growth and connections to matchings, outer space, and moduli-space–style objects such as the pants graph.

Abstract

The present note concerns the "graph of graphs" that has cubic graphs as vertices connected by edges represented by the so-called Whitehead moves. Here, we prove that the outer-conductance of the graph of graphs tends to zero as the number of vertices tends to infinity. This answers a question of K. Rafi in the negative.

Figures (7)

• Figure 1: Two possible Whitehead moves performed on an edge $\varepsilon$.
• Figure 2: The edge $\varepsilon = (v_0, v_1)$ that belongs to a $k$--cycle $C_k$ in $g \in V(\Gamma_n)$ on which a Whitehead move may be performed.
• Figure 3: The resulting $(k-1)$--cycle $C_{k-1}$ in $\tilde{g} = w_\varepsilon(g)$: the edge $\varepsilon = (v, w)$ does not belong to $C_{k-1}$.
• Figure 4: The $k$--cycle $C_k$ ($k \geq 2$) is finally reduced to a loop by $k-1$ consecutive Whitehead moves.
• Figure 5: A $(2, 3)$--lollipop tree

Theorems & Definitions (8)

• Theorem 1: \ref{['wm-thm:graph-of-graphs']}
• Corollary 2: \ref{['cor:graph-of-graphs-1']}
• Definition 2.1
• Theorem 2.2
• Remark 3.1
• Corollary 4.1
• proof
• Remark 4.2