Expansion properties of Whitehead moves on cubic graphs

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.

Keywords

• Trivalent graph
• cubic graph
• expander graph
• Whitehead move
• conductance