Ramanujan Graphs: Optimal Spectral Expanders

• Ramanujan graphs are highly regular finite graphs defined by tight eigenvalue bounds that ensure optimal spectral gap and rapid mixing.
• They are constructed using diverse arithmetic, algebraic, and combinatorial methods, linking classical number theory with modern expander theory.
• Their superior expansion properties support applications ranging from network design and coding theory to quantum computation and combinatorial rigidity.

Ramanujan graphs are highly regular finite graphs exhibiting optimal spectral and expansion properties, deeply connected to number theory, algebra, and combinatorial group theory. Defined as k-regular graphs whose non-trivial eigenvalues lie within the spectrum of the infinite k-regular tree, Ramanujan graphs demonstrate extremal behaviors in mixing times, connectivity, and geometric structure. Their construction and analysis intertwine several mathematical domains, from the Ramanujan conjecture for modular forms to the structure of division algebras, difference sets, and covering theory. These graphs are central to expander theory and have wide-ranging applications in computer science, network design, coding theory, combinatorics, and quantum information.

1. Spectral Definition and Fundamental Properties

A finite, connected, k-regular graph $X$ (with adjacency matrix $A$) is Ramanujan if every eigenvalue $\lambda$ of $A$ other than the trivial values $\pm k$ satisfies

$|\lambda| \le 2 \sqrt{k-1}$

This bound coincides with the spectrum of the adjacency operator on the universal cover of $X$, namely the infinite k-regular tree $T_k$ (Lubotzky, 2017), as shown by Kesten. The optimality of this bound is captured by the Alon–Boppana theorem: for any infinite family of k-regular graphs, $\liminf \lambda_2 \ge 2\sqrt{k-1}$. Thus, Ramanujan graphs maximize the spectral gap $k - \lambda_2$.

Key consequences include:

• Optimal expansion: The Cheeger constant $h(X)$ and $\lambda_1(X)$ are tightly related by

$\frac{h(X)^2}{2k} \le k - \lambda_1(X) \le 2 h(X)$

• Rapid mixing: Random walks on Ramanujan graphs mix to stationarity at the theoretical optimum (cutoff at $t = \frac{d}{d-2}\log_{d-1} n$ up to a $O(\sqrt{\log n})$ window) (Lubetzky et al., 2015, Lubotzky et al., 2019).
• Bisection bandwidth and diameter: Structural properties such as bisection bandwidth $BW(X) \ge \frac{(k-\lambda_2)n}{4}$ and diameter bounds scale optimally with the spectral gap (Aksoy et al., 2019).

2. Constructions: Arithmetic and Combinatorial Techniques

2.1 Number-Theoretic and Algebro-Geometric Methods

The first explicit, infinite families were constructed by Lubotzky, Phillips, and Sarnak, and by Margulis, via quotients of Bruhat–Tits trees for $\operatorname{PGL}_2$ over local fields (quaternion algebras, division algebras, and automorphic representations) (Dahan et al., 2010, Ballantine et al., 2015, Sardari et al., 2019). The LPS graphs are $(p+1)$-regular and constructed from arithmetic lattices acting on trees.

Recent generalizations use:

• Octonion algebras: Utilizing the theory of integral octonions and their unique factorization, new families of Ramanujan graphs with unusually large girth have been constructed. The non-associativity of the octonions is controlled via alternativity and norm multiplicativity, and the Cayley graph is formed from cosets of octonion integral orders using products by octonionic primes. Large girth arises from the absence of factorizational "collisions" (Dahan et al., 2010).
• Ramanujan complexes: Higher-dimensional analogues constructed from quotients of Bruhat–Tits buildings of higher rank p-adic groups. These complexes inherit spectral properties—via the representation theory of Hecke algebras—that generalize Ramanujan expansion to multidimensional settings (Golubev et al., 2014, Lubotzky et al., 2019).

2.2 Algebraic and Combinatorial Constructions

• Difference set and Cayley graph methods: Infinite families for multiple degrees $k$ have been constructed using generalized difference sets in Abelian groups (Cao, 2013). For example, for any even $k>4$ with $2(k+2)=rs$, with $r,s$ even and constrained, a $k$-regular Ramanujan Cayley graph is built on $G = \mathbb{Z}_r \times \mathbb{Z}_s$ with explicit eigenvalue formulas derived from group characters.
• Generalized Ramanujan graphs: The notion extends to covers of arbitrary infinite bounded-degree graphs $X$; a finite quotient $G$ is $X$-Ramanujan if $\lambda_2(G)\leq\rho(X)$, where $\rho(X)$ is the spectral radius (Mohanty et al., 2019). The interlacing polynomials method and new graph polynomials such as the "additive characteristic polynomial" enable broader constructions, including for graphs with local structure richer than trees.

2.3 Polynomial-Time and Randomized Construction Algorithms

• Interlacing families method: The Marcus-Spielman-Srivastava breakthrough established that for every degree and size, there exist bipartite (multi)graphs with the Ramanujan property (Cohen, 2016). Efficient (polynomial-time) algorithms construct these graphs by recursively computing expected characteristic polynomials and traversing an interlacing family/tree using symbolic determinant identities.
• Shift-lift and covering techniques: Forcing larger and more varied Ramanujan graphs via $k$-lifts or shift-lifts. Existence of such lifts preserving the Ramanujan property for $k=2,3,4$ is established, with further efficiency possible for large $k$ (Chandrasekaran et al., 2015, Hall et al., 2015).

3. Expansion, Diameter, and Unique Neighbor Properties

Ramanujan graphs are the prototypical optimal expanders, but their expansion properties admit finer nuances:

• Edge and vertex expansion: While spectral expansion implies strong edge expansion (Cheeger inequality), it does not guarantee unique neighbor or lossless expansion. Explicit counterexamples constructed using the closed orbit method demonstrate that classic number-theoretic Ramanujan graphs such as LPS or Morgenstern graphs are not unique neighbor expanders, falsifying prior conjectures (Kamber et al., 2021).
• Diameter bounds: Classical results gave $\mathrm{diam}(G)\leq 2\log_{k-1} n + O(1)$ for $k$-regular Ramanujan graphs; improvements using analytic number theory (circle method, exponential sums) provide diameter $\leq (\frac{4}{3}+\varepsilon)\log_q |G|$ for certain infinite families, and in some combinatorial constructions the diameter is uniformly bounded by three via iterative line graph and complement operations (Sardari et al., 2019, Ebrahimi, 20 Sep 2024).
• Bipartite and unique neighbor expansions: Novel explicit constructions yield families of bipartite Ramanujan graphs which, when composed with fixed gadgets (through routed product operations), yield bipartite unique-neighbor expanders, underpinned by sharp average degree bounds for induced subgraphs that are valid only for exact Ramanujan graphs (Asherov et al., 2023).

4. Generalizations and Robust Notions

• Powered graphs and generalized Alon–Boppana theorems: For irregular graphs (including Erdős–Rényi and block models), powering ($r$th graph power) smooths local irregularities, yielding spectral gaps close to the theoretical bound $(r+1)(d-1)^{r/2}$, thus supporting a robust, general notion of "Ramanujan-ness" suitable for non-regular graphs (Abbe et al., 2020).
• Ramanujan digraphs and almost-normality: A digraph analogue defines a k-regular digraph as Ramanujan if its nontrivial eigenvalues have $|\lambda|\leq\sqrt k$. In almost-normal (block-diagonalizable) settings, this property is preserved and captures rapid mixing, small diameter, and concentration inequalities even in non-symmetric cases (Parzanchevski, 2018).
• High-dimensional expanders: Ramanujan complexes arising as quotients of higher-rank Bruhat–Tits buildings exhibit concentration of Hodge-Laplace spectra, strong combinatorial expansion of higher dimensional faces (simplices), and refined Cheeger and mixing inequalities (Golubev et al., 2014).

5. Applications: Mixing, Networks, Quantum Computation, and Rigidity

• Optimal mixing and random walks: Ramanujan graphs achieve cutoff for simple and non-backtracking random walks at minimal times, unmatched by general expanders. For quantum computation, their geometric and spectral properties yield "golden gate" sets with optimal covering and compiling properties in compact groups such as $PU(n)$ (Lubetzky et al., 2015, Lubotzky et al., 2019).
• Supercomputing networks: Due to their maximal spectral gap, Ramanujan graphs serve as benchmarks for network design, achieving high bisection bandwidth, low diameter, and robust fault tolerance—properties not matched by traditional architectures like tori or hypercubes (Aksoy et al., 2019).
• Combinatorial rigidity: For sufficiently large order (and, in some cases, under vertex-transitivity), virtually all $k$-regular Ramanujan graphs are globally rigid in $\mathbb{R}^2$ and satisfy other forms of rigidity (body-bar, body-hinge, surfaces), closely tying spectral expansion to geometric stability (Cioabă et al., 2022).
• Enumeration and computational geometry: Randomized growth processes on expander or Ramanujan graphs enable nearly optimal enumeration of large classes of combinatorial structures (e.g., triangulations) via random walks, supported by tight queue and process density bounds derived via spectral gap and expander mixing (Boehm et al., 2019).

6. Limitations, Counterexamples, and Future Directions

Although Ramanujan graphs excel in spectral expansion:

• They do not universally guarantee unique neighbor or lossless expansion; new combinatorial paradigms (e.g., closed orbit method) yield explicit subgraphs of Ramanujan graphs with extremal, poor expansion properties (Kamber et al., 2021).
• Spectral expansion may not imply optimal expansion for all combinatorial or geometric notions, necessitating additional constraints in applications.
• The exact Ramanujan property is crucial in constructions relying on recursive non-backtracking path counts; nearly-Ramanujan graphs may not suffice (Asherov et al., 2023).

Ongoing research explores further extensions:

• Robust spectral expansion in random and block model graphs via powering or alternate operators.
• Expander properties in high dimension, via Ramanujan complexes and their analogues.
• Algorithmic applications leveraging explicit and efficient Ramanujan constructions across computation, communication, and quantum information.

7. Key Formulas and Notational Summary

Property / Object	Formula or Definition
Ramanujan condition	$|\lambda| \le 2\sqrt{k-1}$ for all nontrivial eigenvalues $\lambda$
Spectral gap (regular)	$k - \lambda_2$
Bisection bandwidth	$BW(X) \ge (\mathrm{algebraic\; connect.})\cdot n/4$
Cheeger inequality	$\frac{h(X)^2}{2k} \le k - \lambda_1(X) \le 2 h(X)$
Diameter bound	$$\mathrm{diam}(G) \le 2\lceil \sqrt{2\Delta/\rho_2 \cdot \log_2 n}\rceil$$
Powered Ramanujan thresh.	$\lambda_2(A^{(r)}) \ge (r+1)(d-1)^{r/2}$ (generalized Alon–Boppana; equality for powered Ramanujan)
Bipartite Ramanujan bound	$$|\lambda| \in [\sqrt{d-1} - \sqrt{c-1},\, \sqrt{d-1} + \sqrt{c-1}]$$

The unifying feature is the tight control of nontrivial eigenvalues relative to the underlying universal cover or additive product graph, enabling a spectrum of theoretical and applied advances across mathematics, computer science, and network theory.