Support of Closed Walks and Second Eigenvalue Multiplicity of the Normalized Adjacency Matrix

TL;DR

This paper establishes bounds on the multiplicity of the second eigenvalue of the normalized adjacency matrix in graphs, linking spectral properties to random walk support and eigenvector entries.

Contribution

It introduces new bounds on eigenvalue multiplicity based on random walk support and Perron eigenvector entries, extending spectral graph theory.

Findings

Bound on second eigenvalue multiplicity for general graphs

Bound for simple regular graphs when degree exceeds logarithm

Eigenvalue interval bounds containing the second eigenvalue

Abstract

We show that the multiplicity of the second normalized adjacency matrix eigenvalue of any connected graph of maximum degree $\Delta$ is bounded by $O(n \Delta^{7/5}/\log^{1/5-o(1)}n)$ for any $\Delta$, and by $O(n\log^{1/2}d/\log^{1/4-o(1)}n)$ for simple $d$-regular graphs when $d\ge \log^{1/4}n$. In fact, the same bounds hold for the number of eigenvalues in any interval of width $\lambda_2/\log_\Delta^{1-o(1)}n$ containing the second eigenvalue $\lambda_2$. The main ingredient in the proof is a polynomial (in $k$) lower bound on the typical support of a closed random walk of length $2k$ in any connected graph, which in turn relies on new lower bounds for the entries of the Perron eigenvector of submatrices of the normalized adjacency matrix.