[Paper Review] A new proof of Friedman's second eigenvalue Theorem and its extension to random lifts

Bordenave, Charles|arXiv (Cornell University)|Feb 16, 2015

Graph theory and applications11 citations

TL;DR

This paper presents a new, simplified proof of Friedman's theorem on the spectral gap of random d-regular graphs, showing that the second-largest eigenvalue is at most $2\sqrt{d-1} + o(1)$ with high probability. It extends the method to random lifts of graphs, proving a weak Ramanujan property for non-backtracking eigenvalues by analyzing weighted paths and controlling tangle-induced deviations via operator norm bounds on modified path matrices.

ABSTRACT

It was conjectured by Alon and proved by Friedman that a random $d$-regular graph has nearly the largest possible spectral gap, more precisely, the largest absolute value of the non-trivial eigenvalues of its adjacency matrix is at most $2\sqrt{d-1} +o(1)$ with probability tending to one as the size of the graph tends to infinity. We give a new proof of this statement. We also study related questions on random $n$-lifts of graphs and improve a recent result by Friedman and Kohler.

Motivation & Objective

• To provide a simplified, more transparent proof of Friedman’s second eigenvalue theorem on the spectral gap of random d-regular graphs.
• To extend the spectral gap analysis to random n-lifts of arbitrary base graphs, particularly focusing on non-backtracking matrix eigenvalues.
• To establish a weak Ramanujan property for random lifts by controlling deviations due to rare subgraph structures (tangles).
• To develop a robust method based on high-moment trace estimates and matrix projection to handle non-uniform expectations in random graph spectra.
• To quantify the spectral gap with explicit error terms involving $ (\log \log n / \log n)^2 $, improving on prior probabilistic bounds.

Proposed method

• Reframe the spectral problem using the non-backtracking matrix $ B $ instead of the adjacency matrix $ A $, leveraging the Ihara-Bass formula to relate eigenvalues.
• Introduce a modified matrix $ B^{(\ell)} $ by discarding walks that encounter tangles, ensuring concentration on tangle-free graphs.
• Apply a projection onto the orthogonal complement of the all-ones vector $ \chi $ to isolate non-trivial eigenvalues and bound $ \|B^{(\ell)}\| $ via operator norms.
• Use high-moment trace method: bound $ \mathbb{E}\|C\|^{2m} \leq \mathbb{E}\operatorname{tr}((CC^*)^m) $ for path-weighted matrices $ C $, with $ m \sim \log n / \log \log n $.
• Control contributions from weighted paths of length $ k = 2m\ell $ by partitioning them into isomorphism classes and bounding the number of such classes via graph invariants (vertices, edges, cyclomatic number).
• Establish a deterministic upper bound on the spectral norm using combinatorial enumeration and probabilistic estimates on the configuration model, with careful handling of excess edges and cycling times.

Experimental results

Research questions

• RQ1 Can a simpler, more transparent proof be given for Friedman’s second eigenvalue theorem on random d-regular graphs?
• RQ2 To what extent do tangle-like subgraphs (e.g., complete graphs) distort the expected trace of high powers of the adjacency matrix?
• RQ3 Does the spectral gap of random lifts of a base graph satisfy a weak Ramanujan property, particularly for non-backtracking eigenvalues?
• RQ4 Can the high-moment trace method be adapted to control operator norms of path matrices in the presence of rare, high-impact substructures?
• RQ5 What is the quantitative rate of convergence of the second-largest eigenvalue to the Alon-Boppana bound in random d-regular graphs?

Key findings

• A new proof of Friedman’s theorem is established, showing that for any $ 0 < a < 1 $, there exists $ c > 0 $ such that $ \mathbb{P}(\mu_2 \vee |\mu_n| \geq 2\sqrt{d-1} + c(\log \log n / \log n)^2) \leq n^{-a} $, improving on the qualitative $ o(1) $ bound.
• The method is extended to random lifts of arbitrary graphs, proving that the non-backtracking spectral radius concentrates near $ \sqrt{\rho_1} $, where $ \rho_1 $ is the Perron eigenvalue of the base graph’s non-backtracking matrix.
• For random n-lifts, the non-backtracking eigenvalues satisfy a weak Ramanujan property: $ \lambda_1 \leq \sqrt{\rho_1} + o(1) $ with high probability, under suitable conditions on the lift size.
• The proof controls the contribution of tangled paths via isomorphism class enumeration, showing that the number of such paths is bounded by $ \rho^s (c\ell m)^{16m g + 22m} $, where $ g $ is the cyclomatic number.
• The operator norm of the modified path matrix $ R_k^{(\ell)} $ is shown to satisfy $ \mathbb{E}\|R_k^{(\ell)}\|^{2m} \leq (c\ell m)^{38m} \rho^{2\ell m} $, leading to a high-probability spectral bound.
• The final bound on the spectral radius is $ \|B_n^{(\ell)}\| \leq (\log n)^{15} \rho^{\ell/2} + o(1) $, which implies $ \lambda_1 \leq \sqrt{\rho_1} + o(1) $ when $ \ell \sim \kappa \log_{d-1} n $, $ \kappa < 1/4 $.