Increasingly Many Bounded Eigenvalues of the Graph of Whitehead Moves

TL;DR

This paper studies the eigenvalues of the graph of graphs formed by cubic graphs connected via Whitehead moves, revealing many small eigenvalues that grow with the graph size, indicating significant bottlenecks.

Contribution

It demonstrates that the graph of graphs has an increasing number of small, bounded eigenvalues as the number of vertices grows, extending previous bottleneck findings.

Findings

Number of bounded eigenvalues increases with n

Eigenvalues are significantly smaller than those of random regular graphs

Identifies bottlenecks in the structure of the graph of graphs

Abstract

In this paper, we investigate the eigenvalues of the Laplacian matrix of the "graph of graphs", in which cubic graphs of order n are joined together using Whitehead moves. Our work follows recent results from arXiv:2303.13923 , which discovered a significant "bottleneck" in the graph of graphs. We found that their bottleneck implies an eigenvalue of order at most O(1). In fact, our main contribution is to expand upon this result by showing that the graph of graphs has increasingly many bounded eigenvalues as n increases to infinity. We also show that these eigenvalues are unusually small, in the sense that they are much smaller than the eigenvalues of a random regular graph with an equal number of vertices and a similar degree.