• Corpus ID: 220793357

Support of Closed Walks and Second Eigenvalue Multiplicity of Regular Graphs

  title={Support of Closed Walks and Second Eigenvalue Multiplicity of Regular Graphs},
  author={Theo McKenzie and Peter Michael Reichstein Rasmussen and Nikhil Srivastava},
We show that the multiplicity of the second adjacency matrix eigenvalue of any connected $d-$regular graph is bounded by $O(n d^{7/5}/\log^{1/5-o(1)}n)$ for any $d$, and by $O(n\log^{1/2}d/\log^{1/4-o(1)}n)$ when $d\ge \log^{1/4}n$ and the graph is simple. In fact, the same bounds hold for the number of eigenvalues in any interval of width $\lambda_2/\log_d^{1-o(1)}n$ containing the second eigenvalue $\lambda_2$. The main ingredient in the proof is a polynomial (in $k$) lower bound on the… 
1 Citations

Figures from this paper

Equiangular lines and regular graphs

This paper uses orthogonal projections of matrices with respect to the Frobenius inner product to obtain upper bounds on N α(r) which significantly improve on the only previously known universal bound of Glazyrin and Yu, as well as taking an important step towards determining N α (r) exactly for all r, α.



Eigenvalue multiplicity and volume growth

Let $G$ be a finite group with symmetric generating set $S$, and let $c = \max_{R > 0} |B(2R)|/|B(R)|$ be the doubling constant of the corresponding Cayley graph, where $B(R)$ denotes an $R$-ball in

The Spectral Radius and the Maximum Degree of Irregular Graphs

It is shown that Delta-\lambda_1 is the largest eigenvalue of the adjacency matrix of G, and the effect of adding or removing few edges on the spectral radius of a regular graph is studied.

Many sparse cuts via higher eigenvalues

Here it is proved that for any integer k ∈ [n], there exist ck disjoint subsets S1, ..., Sck, such that [ maxi φ(Si) ≤ C √(λk log k) ] where λk is the kth smallest eigenvalue of the normalized Laplacian and c<1,C>0 are suitable absolute constants.

Eigenvalue multiplicity in regular graphs

Equiangular lines with a fixed angle

Solving a longstanding problem on equiangular lines, we determine, for each given fixed angle and in all sufficiently large dimensions, the maximum number of lines pairwise separated by the given

Multiway Spectral Partitioning and Higher-Order Cheeger Inequalities

This work shows that in every graph there are at least at least 2 disjoint sets, and shows that the √log <i>k</i> bound is tight, up to constant factors, for the “noisy hypercube” graphs.

Eigenvalues and expansion of regular graphs

Improved results on random walks on expanders and selection networks of smaller size than was previously known are obtained and shown to be the best bound one can obtain using the second eigenvalue method.

Decreasing the spectral radius of a graph by link removals.

The minimization of the spectral radius by removing m links is shown to be an NP-complete problem, which suggests considering heuristic strategies, and a scaling law where the decrease in spectral radius is inversely proportional to the number of nodes N in the graph is deduced.

Characterizing graphs of maximum principal ratio

The principal ratio of a connected graph, denoted $\gamma(G)$, is the ratio of the maximum and minimum entries of its first eigenvector. Cioab\u{a} and Gregory conjectured that the graph on $n$