Support of Closed Walks and Second Eigenvalue Multiplicity of the Normalized Adjacency Matrix
Abstract
We show that the multiplicity of the second normalized adjacency matrix eigenvalue of any connected graph of maximum degree $\Delta$ is bounded by $O(n \Delta^{7/5}/\log^{1/5o(1)}n)$ for any $\Delta$, and by $O(n\log^{1/2}d/\log^{1/4o(1)}n)$ 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 $\lambda_2/\log_\Delta^{1o(1)}n$ 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.
 Publication:

arXiv eprints
 Pub Date:
 July 2020
 arXiv:
 arXiv:2007.12819
 Bibcode:
 2020arXiv200712819M
 Keywords:

 Mathematics  Combinatorics;
 Computer Science  Discrete Mathematics;
 Mathematics  Metric Geometry;
 Mathematics  Probability;
 Mathematics  Spectral Theory
 EPrint:
 A previous version of this paper proved the main result for dregular graphs. The current version proves a more general result for the normalized adjacency matrix of bounded degree graphs. 24pp, 3 figures