Increasingly Many Bounded Eigenvalues of the Graph of Whitehead Moves

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.