Near-Ramanujan Graphs: Spectral Properties

• Near-Ramanujan graphs are finite graphs that nearly meet the optimal Ramanujan spectral bound, with almost all eigenvalues contained within the tree spectrum.
• They are constructed using methods such as iterated 2-lifts, interlacing polynomials, and random lifts, ensuring that only a vanishing fraction exceed the threshold.
• Their local tree-likeness, strong spectral rigidity, and controlled eigenvector localization make them crucial in expander theory, combinatorics, and algorithmic applications.

Near-Ramanujan graphs constitute a class of finite graphs whose nontrivial spectral behavior asymptotically approaches, without achieving, the optimality prescribed by the Ramanujan bound. While Ramanujan graphs (in various senses) saturate extremal spectral thresholds governing expansion, mixing time, and combinatorial regularity, near-Ramanujan graphs demonstrate similar extremal behavior up to a small additive relaxation in spectral radius or an overwhelming fraction of “good” eigenvalues. Their construction, spectral properties, and local-global structure occupy a central role in extremal combinatorics, expander theory, and the modern theory of graph limits.

1. Spectral Definitions: Near-Ramanujan Notions

Let $G_n$ be a sequence of finite, connected $k$-regular graphs (or, more generally, graphs with a fixed universal cover $T$). The adjacency spectrum is the multiset of real eigenvalues $\lambda_1(G)\geq \cdots \geq \lambda_N(G)$. The universal cover $T$—typically an infinite, locally finite tree—has adjacency operator $A_T$ with spectrum $\sigma(T)$ and spectral radius $\rho(T)$. In the $k$-regular case, $\rho(T) = 2\sqrt{k-1}$.

A sequence $(G_n)$ is called weakly Ramanujan or near-Ramanujan if, denoting by $$q_n = \#\{i: |\lambda_i(G_n)|\leq\rho(T)\}/|V(G_n)|$$, one has $q_n\rightarrow 1$ as $n\rightarrow \infty$; i.e., almost all eigenvalues are contained in the “tree spectrum” $[-\rho(T),\rho(T)]$ (Huang et al., 2018). This relaxes the constraint that all nontrivial eigenvalues achieve the strict Ramanujan bound, and admits families where a vanishing proportion of “bad” eigenvalues slightly exceed $\rho(T)$.

In the $k$-regular context, one also defines an $\epsilon$–near-Ramanujan graph as a $k$-regular $G$ with $$\lambda(G) = \max\{|\lambda_2|, |\lambda_n|\} \leq 2\sqrt{k-1} + \epsilon$$. Typical results guarantee explicit construction of such graphs for every $(k, \epsilon)$ (Mohanty et al., 2019, O'Donnell et al., 2020).

2. Structural and Local Properties

A pivotal result is that sequences of near-Ramanujan graphs are locally tree-like in a strong sense: If $(G_n)$ shares a common universal cover $T$ and $q_n\to 1$, then for every fixed radius $r$, the proportion of vertices whose $r$-ball is acyclic (a tree) approaches $1$ (Huang et al., 2018). In particular, the essential girth (length of the shortest non-tree-like cycle seen locally) diverges, and the local limit in the Benjamini–Schramm topology converges to $T$.

This asymptotic local tree-likeness is deduced via strong spectral rigidity theorems for sofic and unimodular network limits: If an infinite limit of finite covers has spectral radius matching the universal cover, it must (almost surely) be a tree, precluding persistent cycle density in the limit (Huang et al., 2018). As a corollary, finite graphs with two or more cycles strictly exceed the tree spectral radius, establishing that only trees and unicyclic graphs can be perfectly Ramanujan in the finite case.

3. Constructions and Quantitative Bounds

Explicit constructions of near-Ramanujan graphs now exist for all degrees and various universal covers. In the $k$-regular case, deterministic polynomial-time algorithms produce $k$-regular, $\epsilon$–near-Ramanujan graphs of arbitrary size via iterated 2-lifts, starting from a base graph with large bicycle-free radius and small spectral radius (Mohanty et al., 2019, O'Donnell et al., 2020). This approach generalizes via interlacing polynomials, trace method analysis, and control of the nonbacktracking operator, ensuring that new eigenvalues in covers stay under the threshold $2\sqrt{k-1}+\epsilon$.

Random lifts and $r$-coverings, governed by group-labelings and the framework of complex reflection groups, further yield near-optimal families, with the spectral bounds controlled by matching polynomials and their higher-order analogues. In the bipartite setting, Hall–Puder–Sawin established the existence of infinitely many bipartite Ramanujan graphs of every degree via fully Ramanujan $r$-coverings (Hall et al., 2015).

Near-Ramanujan graphs have also been constructed for generalized universal covers—including free products and additive products of atoms—notably in the $X$-Ramanujan paradigm, where the bound uses the spectral radius $\rho(X)$ of the infinite cover $X$ (Mohanty et al., 2019, O'Donnell et al., 2020).

4. Local Obstructions, Girth, and Eigenvector Structure

While the ℓ₂-spectral criterion alone can guarantee only modest vertex expansion (at least $k/2$ in a $k$-regular graph), local subgraphs (such as copies of $K_{2,k}$ or short cycles) can still force obstructions to expansion in near-Ramanujan graphs (McKenzie et al., 2020). Imposing high girth conditions can remove these local obstructions: for example, in $k$-regular Ramanujan graphs with girth $C\log n$, every set of size $n^{0.99C/4}$ enjoys expansion close to $k$ (McKenzie et al., 2020).

Distinctively, high-girth near-Ramanujan graphs can exhibit localized eigenvectors ("scarring"). There exist infinite sequences of $(d+1)$-regular graphs with near-Ramanujan spectral gap—specifically, $\lambda_2(G)\leq (3/\sqrt{2})\sqrt{d}$—having many eigenvectors fully localized on small sets of $O(|V|^\alpha)$ for $\alpha < 1/6$, demonstrating that optimal expansion and high girth do not imply complete quantum ergodicity or delocalization (Alon et al., 2019).

5. Application to Generalized Graph Classes and Powered Graphs

The concept of near-Ramanujan graphs has been extended to broad settings, including irregular graphs, biregular bipartite graphs (bigraphs), and powered graphs. In the biregular bipartite case, the Ramanujan bound becomes $\sqrt{l-1}+\sqrt{m-1}$, with near-Ramanujan behavior given by a small perturbation of this constant (Ballantine et al., 2015). For $r$-th powers $G^{(r)}$ (where edges connect vertices at distance at most $r$), a generalized Alon–Boppana bound and an $r$-Ramanujan property have been developed, with random regular and Erdős–Rényi graphs shown to be "almost $r$-Ramanujan" after powering—sometimes even more robustly than when measured via nonbacktracking spectrum (Abbe et al., 2020).

The Cayley graph perspective includes explicit near-Ramanujan cases arising from generalized quaternion groups, where the sharpness of the threshold for Ramanujan property and small excesses into the near-Ramanujan regime are connected to deep problems involving primes represented by quadratic polynomials (the Hardy–Littlewood conjecture) (Yamasaki, 2016).

6. Significance, Open Problems, and Future Directions

Near-Ramanujan graphs serve as essential objects in extremal spectral graph theory, expanders, probabilistic combinatorics, and the study of random CSPs. Their local structure, spectral rigidity, and explicit constructibility are central to algorithmic applications requiring optimal or near-optimal spectral gaps. The existence and structural characterizations of near-Ramanujan families illuminate the trade-off between high expansion, large girth, eigenvector localization, and algorithmic implementability.

Open problems include:

• Effective quantitative trade-offs between essential girth, the fraction $q_n$, and the spectral radius excess in weakly Ramanujan sequences (Huang et al., 2018).
• Classification of unimodular networks with minimal spectral radius in generalized covers.
• Pushing "scarring" constants as close as possible to the Ramanujan threshold while preserving eigenvector localization (Alon et al., 2019).
• The determinization of strongly explicit constructions for all degrees, and further reductions in the failure probability or resource requirements in pseudorandom edge-signing methods (Mohanty et al., 2019, O'Donnell et al., 2020).
• Generalization to non-backtracking spectra, higher-dimensional Ramanujan complexes, and spectral bounds with respect to more general infinite covers.

The theory of near-Ramanujan graphs thus occupies a rich boundary between combinatorial, spectral, and probabilistic phenomena in modern graph theory, continuing to motivate advances in group-theoretic constructions, polynomial method analysis, and network theory.